The problem and the result
The exact molecular electronic Hamiltonian has unusually rigid input data: its external potential is generated by a list of positive nuclei. A hardness proof for a matrix obtained by projecting onto supplied orbitals does not by itself establish hardness for this operator. Projection provides an upper bound on the ground energy, while a reduction also needs a lower bound that excludes all omitted continuum states. We give such a reduction for the integer-charge, rational-position encoding below.
The precise promise problem
Fix a positive integer c c c R 1 , … , R M ∈ Q 3 R_1,\ldots,R_M \in\mathbb{Q}^3 R 1 , … , R M ∈ Q 3 Z 1 , … , Z M Z_1,\ldots,Z_M Z 1 , … , Z M N ≥ 1 N \ge1 N ≥ 1 a < b a < b a < b M ≥ 1 M \ge1 M ≥ 1 N N N L L L b − a ≥ L − c b-a \ge L^{-c} b − a ≥ L − c
On ⋀ N L 2 ( R 3 ; C 2 ) \bigwedge^N L^2(\mathbb{R}^3;\mathbb{C}^2) ⋀ N L 2 ( R 3 ; C 2 )
H ( R , Z , N ) = − 1 2 ∑ ℓ = 1 N Δ x ℓ − ∑ ℓ = 1 N ∑ α = 1 M Z α ∣ x ℓ − R α ∣ + ∑ 1 ≤ ℓ < k ≤ N 1 ∣ x ℓ − x k ∣ . (1) H(R,Z,N)=-\frac{1}{2}\sum_{\ell=1}^{N}\Delta_{x_\ell}-\sum_{\ell=1}^{N}\sum_{\alpha=1}^{M}\frac{Z_\alpha}{\lvert x_\ell-R_\alpha\rvert}+\sum_{1\le\ell<k\le N}\frac{1}{\lvert x_\ell-x_k\rvert}. \tag*{(1)} H ( R , Z , N ) = − 2 1 ℓ = 1 ∑ N Δ x ℓ − ℓ = 1 ∑ N α = 1 ∑ M ∣ x ℓ − R α ∣ Z α + 1 ≤ ℓ < k ≤ N ∑ ∣ x ℓ − x k ∣ 1 . (1) The form domain is
Q N = H 1 ( ( R 3 ) N ; ( C 2 ) ⊗ N ) ∩ ⋀ N L 2 ( R 3 ; C 2 ) . \mathcal{Q}_N=H^1\left((\mathbb{R}^3)^N;(\mathbb{C}^2)^{\otimes N}\right)\cap\bigwedge^N L^2(\mathbb{R}^3;\mathbb{C}^2). Q N = H 1 ( ( R 3 ) N ; ( C 2 ) ⊗ N ) ∩ ⋀ N L 2 ( R 3 ; C 2 ) . Antisymmetry exchanges spatial and spin coordinates together. We use the lower-bounded self-adjoint Coulomb operator associated with the closed form on Q N \mathcal{Q}_N Q N
E 0 ( R , Z , N ) = inf Spec H ( R , Z , N ) . E_0(R,Z,N)=\inf\operatorname{Spec} H(R,Z,N). E 0 ( R , Z , N ) = inf Spec H ( R , Z , N ) . The YES promise is E 0 ≤ a E_0 \leq a E 0 ≤ a E 0 ≥ b E_0 \geq b E 0 ≥ b
There is no supplied basis, electron state, magnetic field, or independently adjustable external potential. The minimization includes every total-spin sector. Nuclear repulsion is omitted because the energy convention is electronic and the nuclei are clamped. The problem places no neutrality, fixed-element, or separate nuclear-separation promise on its inputs. In particular, the threshold gap is a precision requirement; it does not assert a molecular spectral gap, a discrete eigenvalue at the infimum, or an efficiently preparable ground state.
Theorem 1.1. The promise problem above is QMA-hard under deterministic classical polynomial-time many-one reductions for c = 1 c = 1 c = 1 b − a ≥ 1 b-a \geq1 b − a ≥ 1
Thus the exact clamped-nuclei problem is QMA-hard for a fixed inverse-polynomial precision, with the input conventions in Table 1 . The binary encoding of unrestricted positive nuclear charges is essential to the construction given here. Charges and inverse nuclear separations may have exponentially large values; their descriptions still have polynomial length. The theorem makes no claim of membership in QMA, or of hardness under an additional bound on nuclear charges.
Table 1. The input model and the precision attained by the reduction. Rational data use the self-delimiting encoding specified in the text.
For clarity, QMA-hardness means that every promise problem admitting polynomial-time quantum verification with a polynomial-size quantum witness reduces to this one by a deterministic classical polynomial-time map preserving YES and NO instances. We use a standard QMA-complete Heisenberg problem as the source. The sole complexity-theoretic input is stated precisely in (2) ; the subsequent construction and its analytic estimates are proved below.
Context and the continuum obstruction
Electronic-structure complexity has several distinct formulations. Liu, Christandl, and Verstraete established QMA-completeness of the finite-mode N N N [5 ] . Restricting the coefficients to those produced by an actual Coulomb potential is a further requirement. Whitfield, Love, and Aspuru-Guzik explained these distinctions and the role of magnetic fields in the early hardness results [16 ] , Sections 3.1 and 6.1.
Schuch and Verstraete proved QMA-completeness for a Hubbard model with local magnetic fields and developed a continuum realization using designed scalar and spin-dependent potentials [13 ] . Their Hubbard-to-Heisenberg route and comparison with the complementary one-particle
space are relevant methodological predecessors. Positive point nuclei impose a different restriction on the external field; the continuum comparison needed here is proved in Section 5.
O’Gorman, Irani, Whitfield, and Fefferman proved QMA-completeness for electronic structure in a supplied finite basis [8 ] , Theorem 1]. Their published Appendix D, Corollary 1, includes positive unit nuclear charges, but the basis remains part of the input; Section V explicitly raises the complete-space question. Indeed, if P P P
inf Spec H ≤ min Spec ( P H P | Ran P ) \inf\operatorname{Spec} H \le\min\operatorname{Spec}\left(PHP\middle|_{\operatorname{Ran} P}\right) inf Spec H ≤ min Spec ( P H P Ran P ) does not transfer a projected NO-instance lower bound to H H H
The Heisenberg hardness theorem of Cubitt, Montanaro, and Piddock [4 , 3 ] supplies a field-free square-lattice source with signed polynomially bounded interactions. Its efficient spatially sparse construction is relevant: generic simulation of a dense interaction graph would not by itself give the required polynomial interaction scales. We use the singlet mediator of Piddock and Montanaro [11 ] to obtain a positive Heisenberg model, and then use repulsive Hubbard superexchange [1 ] . The finite calculations below supply explicit scalar shifts, errors, rationalization, and the geometric control required by the nuclear construction.
Schleich et al. [12 ] study chemically motivated preparation and dynamics through assumptions on accessible chemical processes. Those assumptions define a different computational task from deciding every promised spectral-infimum instance considered here.
The companion article QMA-hardness of continuum Coulomb energy with unit nuclear charges [9 ] , Theorem 1] proves hardness with every Z α = 1 Z_\alpha= 1 Z α = 1
Binding questions are also distinct from the decision problem. For the same kinetic normalization, the companion ionization theorem [10 ] , Theorem 1.1] states that strict binding E N < E N − 1 E_N < E_{N-1} E N < E N − 1 N ≤ Z t o t + C M N \le Z_{\mathrm{tot}} + CM N ≤ Z tot + C M C C C Z t o t = ∑ α Z α Z_{\mathrm{tot}} = \sum_\alpha Z_\alpha Z tot = ∑ α Z α E 0 = 0 E_0 = 0 E 0 = 0
Mechanism of the reduction
We first replace each signed Heisenberg bond by positive interactions with a strongly bound ancillary singlet. A repulsive Hubbard model at half filling then realizes these positive exchanges through virtual double occupancy: its low-energy singly occupied states carry the spins, and the effective exchange is proportional to the square of the hopping strength. The continuum construction must therefore realize prescribed small positive hoppings together with a fixed onsite repulsion. The continuum comparison retains all configurations and spin sectors in the selected orbitals; single occupancy is obtained in the finite Hubbard analysis, not imposed on admissible continuum states.
The construction uses a large integer scale Z ∗ ≍ e D Z_* \asymp e^D Z ∗ ≍ e D D D D y = Z ∗ x y = Z_*x y = Z ∗ x Z ∗ Z_* Z ∗ − 1 / 2 -1/2 − 1/2 Z ∗ Z_* Z ∗ Z ∗ Z_* Z ∗ 5 / 8 5/8 5/8
Three quantitative ingredients make the continuum and bit-complexity requirements compatible.
(i) The sites are placed so that small displacements can independently change every required bond length to first order, with a quantitative bound on the displacement size. A relative tail estimate, proved with capped remote fields and comparison barriers, converts these changes into controlled hopping ratios. Absolute proximity to a hydrogen orbital would not give the required relative precision.
(ii) A graded polynomial Ritz space evaluates a one-well eigenvalue to exponential accuracy in polynomial time. Uniform analytic estimates, an explicit rational mass-matrix lower bound, and certified eigenvalue enclosures make the procedure valid even for skinny cells and ill-conditioned intermediate bases. The unknown eigenfunction is used only to prove approximation, not as an input to the algorithm.
(iii) Localization yields a fixed gap on the entire one-particle complement. A kinetic-trace and Hardy estimate controls Coulomb coupling on every fermionic superposition in the selected modes. Completing a square against the complementary gap gives a lower bound for the full many-electron spectral infimum.
These arguments provide quantitative tools for constructions involving exponentially weak tunneling and a continuum representation. Their uniformity is part of the proof: errors must remain small after multiplication by Z ∗ Z_* Z ∗ [7 ] , with separate rational conditioning and integration bounds. The complement estimate uses spatial localization [14 ] and a quadratic elimination argument whose finite-dimensional analogue is second-order perturbation theory [2 ] . These methods are credited at their points of use; their needed quantitative forms are proved here.
The order of construction and the order of justification deserve separate attention. We first prescribe the finite Hubbard hoppings and compute the primary centers from positive integrals of explicit capped Coulomb fields. These surrogate fields do not involve any unknown orbital or secondary displacement. Only after rounding the primary centers and fixing the integer charges do we tune the secondary nuclei. The surrogate replacement bounds are uniform throughout the tuning cube, and the relative orbital estimates apply to every tuned output. Finally, the continuum lower bound transfers the Hubbard NO promise to arbitrary wavefunctions, and an affine change of energy transfers both promises to rational physical thresholds.
Section 2 develops the finite models and the contact layout. Section 3 proves the isolated-well estimates and the eigenvalue oracle. Section 4 computes the positions and calibrates hopping. Section 5 proves the full continuum comparison. Section 6 chooses all parameters, gives the classical reduction in its actual order of execution, and verifies the final thresholds and encoding.
Throughout, constants denoted by C , c > 0 C,c > 0 C , c > 0
We first reduce a signed spin Hamiltonian to a half-filled Hubbard model. The geometric construction is part of this reduction: every Hubbard hopping will later be set by a small displacement of a nuclear site. For a finite-dimensional self-adjoint operator A A A E ( A ) E(A) E ( A )
h i j s = X i X j + Y i Y j + Z i Z j = σ i ⋅ σ j . h_{ij}^{s}=X_iX_j+Y_iY_j+Z_iZ_j=\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_j . h ij s = X i X j + Y i Y j + Z i Z j = σ i ⋅ σ j . Its eigenvalues are − 3 -3 − 3 1 1 1 ∥ h i j s ∥ = 3 \lVert h_{ij}^{s}\rVert=3 ∥ h ij s ∥ = 3
The source problem and a quantitative perturbation bound
Theorem 2.1 (Source problem). There is a QMA-complete promise problem whose Hamiltonians are
H i n = ∑ e = L R J e h L R s H_{\mathrm{in}}=\sum_{e=LR}J_eh_{LR}^{s} H in = e = L R ∑ J e h L R s on finite, nonwrapping subgraphs of the square lattice, with no one-qubit terms. The signed weights J e J_e J e n ≥ 2 n\ge2 n ≥ 2 a s < b s a_s<b_s a s < b s δ s = b s − a s ≥ 1 / poly ( n ) \delta_s=b_s-a_s\ge1/\operatorname{poly}(n) δ s = b s − a s ≥ 1/ poly ( n )
This is the square-lattice Heisenberg consequence of [3 ] , using the polynomial-weight spatially sparse branch of Lemma 47 there. The Heisenberg interaction has rank-three coupling matrix, so it is outside the exceptional rank-one family in that theorem; the square-lattice construction has no one-qubit fields. The following normalization makes the encoding interface explicit.
Let δ > 0 \delta>0 δ > 0 q s q_s q s ε 0 = δ / 100 \varepsilon_0=\delta/100 ε 0 = δ /100 ε 0 / max { 1 , q s } \varepsilon_0/\max\{1,q_s\} ε 0 / max { 1 , q s } ∥ h i j s ∥ = 3 \lVert h_{ij}^{s}\rVert=3 ∥ h ij s ∥ = 3 3 ε 0 3\varepsilon_0 3 ε 0 c 0 c_0 c 0 a 0 , b 0 a_0,b_0 a 0 , b 0 ε 0 \varepsilon_0 ε 0 c ~ 0 , a ~ 0 , b ~ 0 \widetilde{c}_0,\widetilde{a}_0,\widetilde{b}_0 c 0 , a 0 , b 0
a s = a ~ 0 − c ~ 0 + 5 ε 0 , b s = b ~ 0 − c ~ 0 − 5 ε 0 . a_s=\widetilde{a}_0-\widetilde{c}_0+5\varepsilon_0,\qquad b_s=\widetilde{b}_0-\widetilde{c}_0-5\varepsilon_0. a s = a 0 − c 0 + 5 ε 0 , b s = b 0 − c 0 − 5 ε 0 . The two threshold approximation errors and the Hamiltonian error sum to at most 5 ε 0 5\varepsilon_0 5 ε 0 b s − a s ≥ δ − 12 ε 0 > 0 b_s-a_s\ge\delta-12\varepsilon_0>0 b s − a s ≥ δ − 12 ε 0 > 0 v v v v − 1 v-1 v − 1
This imported source theorem is the only hardness theorem we need. In what follows zero-weight edges are omitted. No lower bound on the magnitudes of the remaining weights is assumed, and no total-spin sector is selected. By appending an isolated original vertex if necessary, at a lattice position outside the existing finite drawing, we may assume that there is at least one original vertex. Tensoring with its uncoupled spin preserves the minimum energy and all polynomial bounds.
We use the following finite-dimensional estimate twice. It controls the bottom energy and does not require a gap within the encoded subspace. The effective second-order operator is the familiar one from Schrieffer–Wolff perturbation theory; see Bravyi, DiVincenzo, and Loss [2 ] .
We prove the particular bound needed here directly, so its constants and the empty-complement case are explicit.
Lemma 2.2 (Second-order bottom-energy estimate). Let A > 0 A > 0 A > 0 P ≠ 0 P \ne0 P = 0 Q = I − P Q = I - P Q = I − P A ≥ g Q A \ge gQ A ≥ g Q g > 0 g > 0 g > 0 B = B ∗ B = B^* B = B ∗ P B P = 0 PBP = 0 P B P = 0 ∥ B ∥ ≤ ϵ g \lVert B\rVert\le\epsilon g ∥ B ∥ ≤ ϵ g 0 ≤ ϵ < 1 / 2 0 \le\epsilon< 1/2 0 ≤ ϵ < 1/2 A − 1 A^{-1} A − 1 P P P
∣ E ( A + B ) − E ( − P B A − 1 B P ∣ P ) ∣ ≤ 2 g ϵ 3 . (2) \left|E(A+B)-E\left(-PBA^{-1}BP\big|_P\right)\right| \le2g\epsilon^3. \tag*{(2)} E ( A + B ) − E ( − P B A − 1 B P P ) ≤ 2 g ϵ 3 . (2) The conclusion also holds when Q = 0 Q = 0 Q = 0
Proof. Let K = P B A − 1 B P ≥ 0 K = PBA^{-1}BP \ge0 K = P B A − 1 B P ≥ 0 κ = ∥ K ∥ \kappa= \lVert K\rVert κ = ∥ K ∥ B p ∈ Q Bp \in Q B p ∈ Q p ∈ P p \in P p ∈ P
0 ≤ κ ≤ ∥ B ∥ 2 g ≤ g ϵ 2 . (3) 0 \le\kappa\le\frac{\lVert B\rVert^2}{g} \le g\epsilon^2. \tag*{(3)} 0 ≤ κ ≤ g ∥ B ∥ 2 ≤ g ϵ 2 . (3) For p ∈ P p \in P p ∈ P q ∈ Q q \in Q q ∈ Q Q B Q ≥ − ϵ g Q ≥ − ϵ A ∣ Q QBQ \ge-\epsilon gQ \ge-\epsilon A|_Q QB Q ≥ − ϵ g Q ≥ − ϵ A ∣ Q
⟨ p + q , ( A + B ) ( p + q ) ⟩ ≥ ( 1 − ϵ ) ⟨ q , A q ⟩ + 2 Re ⟨ q , B p ⟩ = ( 1 − ϵ ) ∥ A 1 / 2 q + A − 1 / 2 B p 1 − ϵ ∥ 2 − ⟨ p , K p ⟩ 1 − ϵ . \begin{aligned}
\langle p+q,(A+B)(p+q)\rangle\ge(1-\epsilon)\langle q,Aq\rangle+ 2\operatorname{Re}\langle q,Bp\rangle\\
&= (1-\epsilon)\left\lVert A^{1/2}q+\frac{A^{-1/2}Bp}{1-\epsilon}\right\rVert^2-\frac{\langle p,Kp\rangle}{1-\epsilon}.
\end{aligned} ⟨ p + q , ( A + B ) ( p + q )⟩ ≥ ( 1 − ϵ ) ⟨ q , A q ⟩ + 2 Re ⟨ q , B p ⟩ = ( 1 − ϵ ) A 1/2 q + 1 − ϵ A − 1/2 B p 2 − 1 − ϵ ⟨ p , K p ⟩ . On a unit vector this is at least − κ / ( 1 − ϵ ) -\kappa/(1-\epsilon) − κ / ( 1 − ϵ )
E ( A + B ) + κ ≥ − ϵ κ 1 − ϵ ≥ − 2 g ϵ 3 . E(A+B)+\kappa\ge-\frac{\epsilon\kappa}{1-\epsilon} \ge-2g\epsilon^3. E ( A + B ) + κ ≥ − 1 − ϵ ϵ κ ≥ − 2 g ϵ 3 . For the other direction choose a unit p ∈ P p \in P p ∈ P K p = κ p Kp=\kappa p K p = κ p q = − A − 1 B p q=-A^{-1}Bp q = − A − 1 B p ∥ q ∥ ≤ ϵ \lVert q\rVert\le\epsilon ∥ q ∥ ≤ ϵ
⟨ p + q , ( A + B ) ( p + q ) ⟩ = − κ + ⟨ q , B q ⟩ . \langle p+q,(A+B)(p+q)\rangle=-\kappa+\langle q,Bq\rangle. ⟨ p + q , ( A + B ) ( p + q )⟩ = − κ + ⟨ q , B q ⟩ . Writing r = ∥ q ∥ 2 r=\lVert q\rVert^2 r = ∥ q ∥ 2
E ( A + B ) + κ ≤ ⟨ q , B q ⟩ + κ r 1 + r ≤ g ϵ 3 + g ϵ 4 ≤ 3 2 g ϵ 3 . E(A+B)+\kappa\le\frac{\langle q,Bq\rangle+\kappa r}{1+r}\le g\epsilon^3+g\epsilon^4\le\frac{3}{2}g\epsilon^3. E ( A + B ) + κ ≤ 1 + r ⟨ q , B q ⟩ + κ r ≤ g ϵ 3 + g ϵ 4 ≤ 2 3 g ϵ 3 . The smallest eigenvalue of − K -K − K P P P − κ -\kappa − κ Q = 0 Q=0 Q = 0 P B P = 0 PBP=0 P B P = 0 B = 0 B=0 B = 0
Replacing signed bonds by positive Heisenberg bonds
Piddock and Montanaro’s singlet mediator construction changes the sign of an effective interaction by coupling to the same or to opposite members of a strongly bound pair [11 ] , Section 2.4 and Lemma 5. We use its isotropic specialization below, keeping the scalar shift and the simultaneous-gadget error explicit. This stage produces only positive spin bonds; the next geometric stage assigns each of them a controllable unit contact.
For every source edge e = L R e=LR e = L R A e A_e A e B e B_e B e k k k
v e = 2 ∣ J e ∣ , C e = { A e , J e < 0 , B e , J e > 0 , v_e=\sqrt{2|J_e|},\qquad C_e=\begin{cases}A_e,&J_e<0,\\ B_e,&J_e>0,\end{cases} v e = 2∣ J e ∣ , C e = { A e , B e , J e < 0 , J e > 0 , H + = Δ ∑ e h A e B e s + Δ ∑ e v e ( h L A e s + h R C e s ) . (4) H_+=\Delta\sum_e h^s_{A_eB_e}+\sqrt{\Delta}\sum_e v_e\left(h^s_{LA_e}+h^s_{RC_e}\right). \tag*{(4)} H + = Δ e ∑ h A e B e s + Δ e ∑ v e ( h L A e s + h R C e s ) . (4) Every nonzero coefficient in H + H_+ H +
V = ∑ e v e , C 0 = 3 k Δ + 3 2 ∑ e v e 2 . (5) V = \sum_e v_e,\qquad C_0 = 3k\Delta+ \frac{3}{2}\sum_e v_e^2. \tag*{(5)} V = e ∑ v e , C 0 = 3 k Δ + 2 3 e ∑ v e 2 . (5) Proposition 2.3 (Positive-spin reduction). If Δ > 3 V \sqrt{\Delta} > 3V Δ > 3 V
∣ E ( H + ) − E ( H i n ) + C 0 ∣ ≤ 27 V 3 Δ . (6) \left| E(H_+) - E(H_{\mathrm{in}}) + C_0 \right| \le\frac{27V^3}{\sqrt{\Delta}}. \tag*{(6)} ∣ E ( H + ) − E ( H in ) + C 0 ∣ ≤ Δ 27 V 3 . (6) In particular, a polynomially large integer Δ \Delta Δ
Proof. Take
A = Δ ∑ e ( h A e B e s + 3 I ) , B = Δ ∑ e v e ( h L A e s + h R C e s ) . A = \Delta\sum_e \left(h^s_{A_eB_e} + 3I\right),\qquad B = \sqrt{\Delta}\sum_e v_e\left(h^s_{LA_e} + h^s_{R C_e}\right). A = Δ e ∑ ( h A e B e s + 3 I ) , B = Δ e ∑ v e ( h L A e s + h R C e s ) . The kernel of A A A P s P_s P s g = 4 Δ g = 4\Delta g = 4Δ
P s B P s = 0 , ∥ B ∥ ≤ 6 Δ V . P_s B P_s = 0,\qquad\lVert B\rVert\le6\sqrt{\Delta}V. P s B P s = 0 , ∥ B ∥ ≤ 6 Δ V . A weak interaction from pair e e e P s P_s P s 4 Δ 4\Delta 4Δ f ≠ e f \ne e f = e e e e
For a singlet ∣ s ⟩ |s\rangle ∣ s ⟩ A , B A,B A , B μ , ν \mu,\nu μ , ν
⟨ s ∣ σ A μ σ A ν ∣ s ⟩ = δ μ ν , ⟨ s ∣ σ A μ σ B ν ∣ s ⟩ = − δ μ ν . \langle s|\sigma_A^\mu\sigma_A^\nu|s\rangle= \delta_{\mu\nu},\qquad\langle s|\sigma_A^\mu\sigma_B^\nu|s\rangle= -\delta_{\mu\nu}. ⟨ s ∣ σ A μ σ A ν ∣ s ⟩ = δ μν , ⟨ s ∣ σ A μ σ B ν ∣ s ⟩ = − δ μν . Consequently the squares of the two weak terms each compress to 3 I 3I 3 I 2 h L R s 2h^s_{LR} 2 h L R s C e = A e C_e=A_e C e = A e − 2 h L R s -2h^s_{LR} − 2 h L R s C e = B e C_e=B_e C e = B e e e e − P s B A − 1 B P s -P_sBA^{-1}BP_s − P s B A − 1 B P s
− v e 2 4 { 6 I + 2 h L R s , J e < 0 , 6 I − 2 h L R s , J e > 0 = J e h L R s − 3 2 v e 2 I . (7) -\frac{v_e^2}{4}
\begin{cases}
6I + 2h^s_{LR}, & J_e < 0,\\
6I - 2h^s_{LR}, & J_e > 0
\end{cases}
= J_eh^s_{LR} - \frac{3}{2}v_e^2I. \tag*{(7)} − 4 v e 2 { 6 I + 2 h L R s , 6 I − 2 h L R s , J e < 0 , J e > 0 = J e h L R s − 2 3 v e 2 I . (7) The factor 4 4 4 Δ \Delta Δ ϵ = 3 V / ( 2 Δ ) < 1 / 2 \epsilon= 3V/(2\sqrt{\Delta}) < 1/2 ϵ = 3 V / ( 2 Δ ) < 1/2 2 ( 4 Δ ) ϵ 3 = 27 V 3 / Δ 2(4\Delta)\epsilon^3 = 27V^3/\sqrt{\Delta} 2 ( 4Δ ) ϵ 3 = 27 V 3 / Δ A + B = H + + 3 k Δ I A+B=H_+ + 3k\Delta I A + B = H + + 3 k Δ I (7) gives the result. If k = 0 k=0 k = 0
Here is one explicit scale choice, which we retain for later bookkeeping. Set
γ = min { 1 , δ s } , W = max ( { 1 } ∪ { ∣ J e ∣ : e } ) , \gamma= \min\{1,\delta_s\},\qquad W = \max\left(\{1\}\cup\{|J_e|:e\}\right), γ = min { 1 , δ s } , W = max ( { 1 } ∪ { ∣ J e ∣ : e } ) , V ‾ = max { 1 , 2 k W } , Δ = ⌈ ( 1728 V ‾ 3 γ ) 2 ⌉ . (8) \overline{V} = \max\{1,2kW\},\qquad\Delta= \left\lceil\left(\frac{1728\overline{V}^3}{\gamma}\right)^2\right\rceil. \tag*{(8)} V = max { 1 , 2 k W } , Δ = ( γ 1728 V 3 ) 2 . (8) Then V ≤ V ‾ V \le\overline{V} V ≤ V (6) is at most γ / 64 \gamma/64 γ /64 v e ≤ Δ v_e \le\sqrt{\Delta} v e ≤ Δ H + H_+ H + Δ \Delta Δ n n n
Place the original lattice vertices at spacing 2 in the x , y x,y x , y x x x y y y ( ℓ , ζ ) (\ell,\zeta) ( ℓ , ζ ) ℓ \ell ℓ ζ = z \zeta=z ζ = z ζ = − z \zeta=-z ζ = − z
A e B e J e < 0 ( 1 , 0 ) ( 1 , 1 ) J e > 0 ( 1 / 2 , 3 / 2 ) ( 3 / 2 , 3 / 2 ) . \begin{equation}
\begin{array}{c|cc}
& A_e & B_e\\ \hline
J_e<0 &(1,0)&(1,1)\\[2pt]
J_e>0 &(1/2,\sqrt3/2)&(3/2,\sqrt3/2).
\end{array}
\end{equation} J e < 0 J e > 0 A e ( 1 , 0 ) ( 1/2 , 3 /2 ) B e ( 1 , 1 ) ( 3/2 , 3 /2 ) . The local endpoints are always L = ( 0 , 0 ) L=(0,0) L = ( 0 , 0 ) R = ( 2 , 0 ) R=(2,0) R = ( 2 , 0 ) p 1 , … , p m p_1,\ldots,p_m p 1 , … , p m Q ( 3 ) 3 \mathbb{Q}(\sqrt{3})^3 Q ( 3 ) 3 Figure 1 shows these placements and the additional contact between consecutive positive gadgets.
Figure 1. The contact construction in the local ( ℓ , ζ ) (\ell,\zeta) ( ℓ , ζ ) h = 3 / 2 h=\sqrt{3}/2 h = 3 /2 w e = Δ v e w_e=\sqrt{\Delta}\,v_e w e = Δ v e Δ \Delta Δ H + H_+ H + ζ = z \zeta=z ζ = z ζ = − z \zeta=-z ζ = − z
Lemma 2.4 (Contact classification). All the sites p i p_i p i 2 \sqrt{2} 2 C C C
L A e LA_e L A e A e R A_eR A e R A e B e A_eB_e A e B e
L A e LA_e L A e A e B e A_eB_e A e B e B e R B_eR B e R
B e A f B_eA_f B e A f e , f e,f e , f
Thus every interaction of H + H_+ H + K i j ≥ 0 K_{ij}\geq0 K ij ≥ 0 K i j = 0 K_{ij}=0 K ij = 0
Proof. Within one gadget the assertion follows directly from (2.8). For a negative gadget the two distances from B e B_e B e 2 \sqrt{2} 2 3 \sqrt{3} 3 3 / 2 3/2 3/2
For ancillas belonging to distinct gadgets on the same axis line, the only along-line separation below 3 / 2 3/2 3/2 B e B_e B e A f A_f A f 3 / 2 > 2 3/2>\sqrt{2} 3/2 > 2
It remains to compare horizontal and vertical ancillas. Call an ancilla midpoint type if it belongs to a negative gadget, and positive type otherwise. Two midpoint-type projections differ by at least 1 1 1 2 \sqrt{2} 2 1 1 1 1 / 2 1/2 1/2 3 / 2 3/2 3/2 1 + 1 / 4 + 3 / 4 = 2 1+1/4+3/4=2 1 + 1/4 + 3/4 = 2 3 \sqrt{3} 3 2 \sqrt{2} 2
Orient each contact arbitrarily and write u i j = p j − p i u_{ij}=p_j-p_i u ij = p j − p i ∣ u i j ∣ = 1 \lvert u_{ij}\rvert=1 ∣ u ij ∣ = 1 s i s_i s i u i j ⋅ ( s j − s i ) u_{ij}\cdot(s_j-s_i) u ij ⋅ ( s j − s i )
Lemma 2.5 (A bounded right inverse for contact lengths). For every real assignment ( d i j ) { i , j } ∈ C (d_{ij})_{\{i,j\}\in C} ( d ij ) { i , j } ∈ C s i ∈ R 3 s_i\in\mathbb{R}^3 s i ∈ R 3
u i j ⋅ ( s j − s i ) = d i j , max i ∣ s i ∣ ≤ 6 m max { i , j } ∈ C ∣ d i j ∣ . (9) u_{ij}\cdot(s_j-s_i)=d_{ij},\qquad\max_i\lvert s_i\rvert\le6m\max_{\{i,j\}\in C}\lvert d_{ij}\rvert. \tag*{(9)} u ij ⋅ ( s j − s i ) = d ij , i max ∣ s i ∣ ≤ 6 m { i , j } ∈ C max ∣ d ij ∣ . (9) For an empty contact set take all displacements zero. The construction is linear in d d d Q ( 3 ) \mathbb{Q}(\sqrt{3}) Q ( 3 )
Proof. Set the height displacement of every original vertex to zero. The horizontal coordinate of each original vertex will be determined using only its original row, and its vertical planar coordinate using only its original column. These are independent variables even when both directions meet at the same vertex.
Fix one such axis line. Write a a a z z z σ = 1 \sigma=1 σ = 1 σ = − 1 \sigma=-1 σ = − 1 a a a
a R = a L + d L A e + d A e R , a A e = a L + d L A e , a B e = a A e , z A e = 0 , z B e = σ d A e B e . (10) \begin{aligned}
a_R&=a_L+d_{LA_e}+d_{A_eR},&\qquad a_{A_e}&=a_L+d_{LA_e},\\
a_{B_e}&=a_{A_e},& z_{A_e}&=0,& z_{B_e}&=\sigma d_{A_eB_e}. \tag*{(10)}
\end{aligned} a R a B e = a L + d L A e + d A e R , = a A e , a A e z A e = a L + d L A e , = 0 , z B e = σ d A e B e . (10) Unconstrained original coordinates may be set to zero. A positive or missing bond places no requirement on the difference of its original endpoints’ a a a (10) satisfies all three constraints in its negative gadgets.
Now consider the elevated vertices of positive gadgets on this line. Their same-height contacts, including the extra contacts of Lemma 2.4, form disjoint paths. Set the first a a a a n e x t = a c u r r e n t + d e d g e a_{\mathrm{next}}=a_{\mathrm{current}}+d_{\mathrm{edge}} a next = a current + d edge a a a h 0 = 3 / 2 h_0=\sqrt{3}/2 h 0 = 3 /2
z A e = d L A e − ( a A e − a L ) / 2 σ h 0 , z B e = d B e R − ( a R − a B e ) / 2 σ h 0 . (11) z_{A_e}=\frac{d_{LA_e}-(a_{A_e}-a_L)/2}{\sigma h_0},
\qquad
z_{B_e}=\frac{d_{B_eR}-(a_R-a_{B_e})/2}{\sigma h_0}. \tag*{(11)} z A e = σ h 0 d L A e − ( a A e − a L ) /2 , z B e = σ h 0 d B e R − ( a R − a B e ) /2 . (11) Indeed the two unit spoke vectors are 1 2 e a x i s + σ h 0 e z \frac{1}{2}e_{\mathrm{axis}}+\sigma h_0e_z 2 1 e axis + σ h 0 e z 1 2 e a x i s − σ h 0 e z \frac{1}{2}e_{\mathrm{axis}}-\sigma h_0e_z 2 1 e axis − σ h 0 e z
Repeat on all rows and columns. The row procedure changes only original x x x y y y
For the norm bound put d max = max C ∣ d i j ∣ d_{\max}=\max_{C}|d_{ij}| d m a x = max C ∣ d ij ∣ 2 m d max 2md_{\max} 2 m d m a x m d max md_{\max} m d m a x ( 2 m + 1 ) d max (2m+1)d_{\max} ( 2 m + 1 ) d m a x d max d_{\max} d m a x (11) , positive ancilla heights are at most ( 1 + 3 m / 2 ) d max / h 0 (1+3m/2)d_{\max}/h_0 ( 1 + 3 m /2 ) d m a x / h 0 6 m d max 6md_{\max} 6 m d m a x 1 / h 0 1/h_0 1/ h 0
Only approximate length equations will be needed. To make their bit complexity explicit, suppose the data are known within τ / 4 \tau/4 τ /4 D p i Dp_i D p i X i X_i X i τ / 4 \tau/4 τ /4 s i = X i − D p i s_i=X_i-Dp_i s i = X i − D p i p i p_i p i τ / 4 + 2 ( τ / 4 ) < τ \tau/4+2(\tau/4)<\tau τ /4 + 2 ( τ /4 ) < τ
max i ∣ s i ∣ ≤ 6 m ( max C ∣ d i j ∣ + τ ) + τ . (12) \max_i |s_i| \le6m\left(\max_{C}|d_{ij}|+\tau\right)+\tau. \tag*{(12)} i max ∣ s i ∣ ≤ 6 m ( C max ∣ d ij ∣ + τ ) + τ . (12) The number of arithmetic operations is polynomial, and all accumulated coefficients have polynomial size. Computing 3 \sqrt{3} 3
For later use, actual distances satisfy the exact vector identities and the elementary estimates
X j − X i = D u i j + s j − s i ( { i , j } ∈ C ) , ∣ X j − X i ∣ = D + u i j ⋅ ( s j − s i ) + O ( S 2 / D ) ( { i , j } ∈ C ) , ∣ X j − X i ∣ ≥ 2 D − 2 S ( { i , j } ∉ C ) . (13) \begin{aligned}
X_j-X_i &= Du_{ij}+s_j-s_i &&(\{i,j\}\in C),\\
|X_j-X_i| &= D+u_{ij}\cdot(s_j-s_i)+O(S^2/D) &&(\{i,j\}\in C),\\
|X_j-X_i| &\ge\sqrt{2}D-2S &&(\{i,j\}\notin C). \tag*{(13)}
\end{aligned} X j − X i ∣ X j − X i ∣ ∣ X j − X i ∣ = D u ij + s j − s i = D + u ij ⋅ ( s j − s i ) + O ( S 2 / D ) ≥ 2 D − 2 S ({ i , j } ∈ C ) , ({ i , j } ∈ C ) , ({ i , j } ∈ / C ) . (13) provided max i ∣ s i ∣ ≤ S \max_i|s_i|\le S max i ∣ s i ∣ ≤ S S ≤ D / 4 S\le D/4 S ≤ D /4
The half-filled Hubbard model
We now realize those positive bonds as second-order hopping processes. Virtual double occupancy gives the antiferromagnetic superexchange mechanism of Anderson [1 ] . In our Pauli-matrix normalization the coefficient is t 2 / U t^2/U t 2 / U h i s − I h_i^s-I h i s − I
Associate two fermionic modes, labelled ↑ , ↓ \uparrow,\downarrow ↑ , ↓ m m m m m m c i σ † , c i σ c_{i\sigma}^{\dagger},c_{i\sigma} c iσ † , c iσ n i σ = c i σ † c i σ n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma} n iσ = c iσ † c iσ
H H u b ( t ) = − ∑ { i , j } ∈ C ∑ σ = ↑ , ↓ t i j ( c i σ † c j σ + c j σ † c i σ ) + U ∑ i n i ↑ n i ↓ . (14) H_{\mathrm{Hub}}(t)=-\sum_{\{i,j\}\in C}\sum_{\sigma=\uparrow,\downarrow}t_{ij}\left(c_{i\sigma}^{\dagger}c_{j\sigma}+c_{j\sigma}^{\dagger}c_{i\sigma}\right)+U\sum_i n_{i\uparrow}n_{i\downarrow}. \tag*{(14)} H Hub ( t ) = − { i , j } ∈ C ∑ σ =↑ , ↓ ∑ t ij ( c iσ † c j σ + c j σ † c iσ ) + U i ∑ n i ↑ n i ↓ . (14) U = ∬ R 3 × R 3 ϕ ( y ) 2 ϕ ( y ′ ) 2 ∣ y − y ′ ∣ d y d y ′ , ϕ ( y ) = π − 1 / 2 e − ∣ y ∣ . U=\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\phi(y)^2\phi(y')^2}{\lvert y-y'\rvert}\,\mathrm{d}y\,\mathrm{d}y',\qquad\phi(y)=\pi^{-1/2}e^{-\lvert y\rvert}. U = ∬ R 3 × R 3 ∣ y − y ′ ∣ ϕ ( y ) 2 ϕ ( y ′ ) 2 d y d y ′ , ϕ ( y ) = π − 1/2 e − ∣ y ∣ . The radial probability density of ϕ 2 \phi^2 ϕ 2 p ( r ) = 4 r 2 e − 2 r p(r)=4r^2e^{-2r} p ( r ) = 4 r 2 e − 2 r 1 / max ( r , s ) 1/\max(r,s) 1/ max ( r , s ) r , s > 0 r,s>0 r , s > 0
1 2 ∫ − 1 1 d u r 2 + s 2 − 2 r s u = 1 max ( r , s ) . \frac{1}{2}\int_{-1}^{1}\frac{\mathrm{d}u}{\sqrt{r^2+s^2-2rsu}}=\frac{1}{\max(r,s)}. 2 1 ∫ − 1 1 r 2 + s 2 − 2 r s u d u = max ( r , s ) 1 . Since ∫ 0 r p ( s ) d s = 1 − e − 2 r ( 1 + 2 r + 2 r 2 ) \int_0^r p(s)\,\mathrm{d}s=1-e^{-2r}(1+2r+2r^2) ∫ 0 r p ( s ) d s = 1 − e − 2 r ( 1 + 2 r + 2 r 2 )
U = 8 ∫ 0 ∞ r e − 2 r [ 1 − e − 2 r ( 1 + 2 r + 2 r 2 ) ] d r = 8 ( 1 4 − 1 16 − 1 16 − 3 64 ) = 5 8 . (15) \begin{aligned}
U&=8\int_0^\infty re^{-2r}\left[1-e^{-2r}(1+2r+2r^2)\right]\,\mathrm{d}r\\
&=8\left(\frac{1}{4}-\frac{1}{16}-\frac{1}{16}-\frac{3}{64}\right)=\frac{5}{8}. \tag*{(15)}
\end{aligned} U = 8 ∫ 0 ∞ r e − 2 r [ 1 − e − 2 r ( 1 + 2 r + 2 r 2 ) ] d r = 8 ( 4 1 − 16 1 − 16 1 − 64 3 ) = 8 5 . (15) Thus U U U
Let q C = ∣ C ∣ q_C=\lvert C\rvert q C = ∣ C ∣
R ‾ = max { 1 , q C Δ } , η = U γ 1024 R ‾ 3 , α = η 2 U , f = α γ 128 max { 1 , q C } . (16) \overline{R}=\max\{1,q_C\Delta\},\qquad\eta=\frac{U\gamma}{1024\overline{R}^{3}},\qquad\alpha=\frac{\eta^2}{U},\qquad f=\frac{\alpha\gamma}{128\max\{1,q_C\}}. \tag*{(16)} R = max { 1 , q C Δ } , η = 1024 R 3 U γ , α = U η 2 , f = 128 max { 1 , q C } α γ . (16) The ideal hoppings are t i j ( 0 ) = η K i j t_{ij}^{(0)}=\eta\sqrt{K_{ij}} t ij ( 0 ) = η K ij r i j r_{ij} r ij ∣ r i j − t i j ( 0 ) ∣ ≤ f \lvert r_{ij}-t_{ij}^{(0)}\rvert\le f ∣ r ij − t ij ( 0 ) ∣ ≤ f
t i j = max { f , r i j } . t_{ij}=\max\{f,r_{ij}\}. t ij = max { f , r ij } . Polynomial-precision bisection on nonnegative square roots (at most two nested square roots here) suffices to compute r i j r_{ij} r ij (2.17) is a comparison of rationals. All final hoppings satisfy 1 / poly ( n ) ≤ t i j < 1 1/\operatorname{poly}(n)\le t_{ij}<1 1/ poly ( n ) ≤ t ij < 1 K i j = 0 K_{ij}=0 K ij = 0
Proposition 2.6 (Hubbard reduction with rational hoppings). The construction above is deterministic and polynomial time. With
C s = C 0 + ∑ { i , j } ∈ C K i j = 3 k Δ + 3 2 ∑ e v e 2 + ∑ { i , j } ∈ C K i j , (17) C_s=C_0+\sum_{\{i,j\}\in C}K_{ij}=3k\Delta+\frac{3}{2}\sum_e v_e^2+\sum_{\{i,j\}\in C}K_{ij}, \tag*{(17)} C s = C 0 + { i , j } ∈ C ∑ K ij = 3 k Δ + 2 3 e ∑ v e 2 + { i , j } ∈ C ∑ K ij , (17) it satisfies
∣ E ( H H u b ( t ) ) − α ( E ( H i n ) − C s ) ∣ ≤ 3 α γ 64 . (18) \left|E(H_{\mathrm{Hub}}(t))-\alpha\left(E(H_{\mathrm{in}})-C_s\right)\right|\le\frac{3\alpha\gamma}{64}. \tag*{(18)} ∣ E ( H Hub ( t )) − α ( E ( H in ) − C s ) ∣ ≤ 64 3 α γ . (18) The scalar C s C_s C s
Proof. First use the ideal hoppings t ( 0 ) t^{(0)} t ( 0 ) A U = U ∑ i n i ↑ n i ↓ A_U=U\sum_i n_{i\uparrow}n_{i\downarrow} A U = U ∑ i n i ↑ n i ↓ T T T H H u b ( t ( 0 ) ) H_{\mathrm{Hub}}(t^{(0)}) H Hub ( t ( 0 ) ) m m m A U A_U A U m m m m m m m m m P s o P_{\mathrm{so}} P so U U U P s o P_{\mathrm{so}} P so U U U P s o T P s o = 0 P_{\mathrm{so}} T P_{\mathrm{so}}=0 P so T P so = 0
A second hop can return to single occupancy only by moving an electron from that doubly occupied site to that vacant site. Thus the two hops must use the same unordered bond; products from different bonds have zero compression, even when the bonds share a vertex. For completeness, consider one bond i , j i,j i , j t t t i ↑ , i ↓ , j ↑ , j ↓ i\uparrow,i\downarrow,j\uparrow,j\downarrow i ↑ , i ↓ , j ↑ , j ↓ ∣ D i ⟩ = c i ↑ † c i ↓ † ∣ 0 ⟩ |D_i\rangle=c_{i\uparrow}^{\dagger}c_{i\downarrow}^{\dagger}|0\rangle ∣ D i ⟩ = c i ↑ † c i ↓ † ∣0 ⟩ ∣ D j ⟩ |D_j\rangle ∣ D j ⟩ T i j T_{ij} T ij
T i j ∣ ↑ , ↓ ⟩ = − t ( ∣ D i ⟩ + ∣ D j ⟩ ) , T i j ∣ ↓ , ↑ ⟩ = t ( ∣ D i ⟩ + ∣ D j ⟩ ) . T_{ij}|\uparrow,\downarrow\rangle=-t(|D_i\rangle+|D_j\rangle),\qquad T_{ij}|\downarrow,\uparrow\rangle=t(|D_i\rangle+|D_j\rangle). T ij ∣ ↑ , ↓ ⟩ = − t ( ∣ D i ⟩ + ∣ D j ⟩) , T ij ∣ ↓ , ↑ ⟩ = t ( ∣ D i ⟩ + ∣ D j ⟩) . It consequently annihilates all triplets and maps the singlet to − 2 t ( ∣ D i ⟩ + ∣ D j ⟩ ) -\sqrt{2}t(|D_i\rangle+|D_j\rangle) − 2 t ( ∣ D i ⟩ + ∣ D j ⟩) 4 t 2 4t^2 4 t 2 i i i j j j
− P s o T i j A U − 1 T i j P s o = − 4 t 2 U P s i n g l e t , i j = t 2 U ( h i j s − I ) . (19) -P_{\mathrm{so}}T_{ij}A_U^{-1}T_{ij}P_{\mathrm{so}}=-\frac{4t^2}{U}P_{\mathrm{singlet},ij}=\frac{t^2}{U}(h_{ij}^{s}-I). \tag*{(19)} − P so T ij A U − 1 T ij P so = − U 4 t 2 P singlet , ij = U t 2 ( h ij s − I ) . (19) Here P s i n g l e t , i j = ( I − h i j s ) / 4 P_{\mathrm{singlet},ij}=(I-h_{ij}^{s})/4 P singlet , ij = ( I − h ij s ) /4
For each spin, c i σ † c j σ + c j σ † c i σ c_{i\sigma}^{\dagger}c_{j\sigma}+c_{j\sigma}^{\dagger}c_{i\sigma} c iσ † c j σ + c j σ † c iσ
∥ T ∥ ≤ 2 ∑ c t i j ( 0 ) = 2 η ∑ c K i j ≤ 2 η R ‾ . \lVert T\rVert\le2\sum_c t_{ij}^{(0)}=2\eta\sum_c\sqrt{K_{ij}}\le2\eta\overline{R}. ∥ T ∥ ≤ 2 c ∑ t ij ( 0 ) = 2 η c ∑ K ij ≤ 2 η R . The last inequality uses K i j ≤ Δ K_{ij}\le\Delta K ij ≤ Δ Δ ≥ 1 \Delta\ge1 Δ ≥ 1 g = U g=U g = U ϵ = 2 η R ‾ / U < 1 / 2 \epsilon=2\eta\overline{R}/U<1/2 ϵ = 2 η R / U < 1/2
∣ E ( H H u b ( t ( 0 ) ) ) − α ( E ( H + ) − ∑ c K i j ) ∣ ≤ 16 η 3 R ‾ 3 U 2 ≤ α γ 64 . (20) \left|E\left(H_{\mathrm{Hub}}(t^{(0)})\right)-\alpha\left(E(H_+)-\sum_c K_{ij}\right)\right|\le\frac{16\eta^3\overline{R}^{3}}{U^2}\le\frac{\alpha\gamma}{64}. \tag*{(20)} E ( H Hub ( t ( 0 ) ) ) − α ( E ( H + ) − c ∑ K ij ) ≤ U 2 16 η 3 R 3 ≤ 64 α γ . (20) If the complementary subspace is empty, as can happen for m = 1 m=1 m = 1
For the rational hoppings, nonnegativity of t i j ( 0 ) t_{ij}^{(0)} t ij ( 0 ) r i j r_{ij} r ij ∣ max { f , r i j } − t i j ( 0 ) ∣ ≤ f |\max\{f,r_{ij}\}-t_{ij}^{(0)}|\le f ∣ max { f , r ij } − t ij ( 0 ) ∣ ≤ f
∥ H H u b ( t ) − H H u b ( t ( 0 ) ) ∥ ≤ 2 q C f ≤ α γ 64 . (21) \left\|H_{\mathrm{Hub}}(t)-H_{\mathrm{Hub}}(t^{(0)})\right\|\le2qCf\le\frac{\alpha\gamma}{64}. \tag*{(21)} H Hub ( t ) − H Hub ( t ( 0 ) ) ≤ 2 q C f ≤ 64 α γ . (21) Also t i j ( 0 ) ≤ η R ‾ ≤ U / 1024 t_{ij}^{(0)}\le\eta\overline{R}\le U/1024 t ij ( 0 ) ≤ η R ≤ U /1024 f ≤ U / ( 128 ⋅ 1024 2 ) f\le U/(128\cdot1024^2) f ≤ U / ( 128 ⋅ 102 4 2 ) f ≤ t i j < 1 f\le t_{ij}<1 f ≤ t ij < 1 f f f f f f C C C
Finally, Proposition 2.3 with (2.7) contributes at most another α γ / 64 \alpha\gamma/64 α γ /64 n n n K i j K_{ij} K ij Δ \Delta Δ 2 Δ ∣ J e ∣ 2\Delta|J_e| 2Δ∣ J e ∣
Isolated wells and deterministic energy tuning
This section supplies an exponentially accurate classical procedure for tuning the individual well energies. The procedure operates on one-electron operators only; it does not diagonalize any of the spin or Hubbard Hamiltonians from Section 2.
There are two separate tasks. First we must show that moving a secondary positive nucleus changes one isolated energy monotonically, with much weaker effects on the other sites. Then we must evaluate those energies accurately enough to implement the resulting contraction. The required accuracy is exponential in D D D
Fix centers X 1 , … , X m ∈ R 3 X_1,\ldots,X_m \in\mathbb{R}^3 X 1 , … , X m ∈ R 3
∣ X i − X j ∣ ≥ D − 2 S ( i ≠ j ) , S ≤ D / 100. (22) \lvert X_i-X_j\rvert\ge D-2S \quad(i\ne j), \qquad S\le D/100. \tag*{(22)} ∣ X i − X j ∣ ≥ D − 2 S ( i = j ) , S ≤ D /100. (22) Here D D D m / D m/D m / D ρ i ∈ [ 1 , 4 ] \rho_i \in[1,4] ρ i ∈ [ 1 , 4 ]
Choose an integer Z ∗ Z_* Z ∗ e D / 2 ≤ Z ∗ ≤ 2 e D e^D/2 \le Z_* \le2e^D e D /2 ≤ Z ∗ ≤ 2 e D
Z ∗ q = round ( Z ∗ ( 1 − 500 m D ) ) , Z_*q=\operatorname{round}\left(Z_*\left(1-\frac{500m}{D}\right)\right), Z ∗ q = round ( Z ∗ ( 1 − D 500 m ) ) , Z ∗ q ′ = round ( Z ∗ 1000 m D ) . (23) Z_*q'=\operatorname{round}\left(Z_*\frac{1000m}{D}\right). \tag*{(23)} Z ∗ q ′ = round ( Z ∗ D 1000 m ) . (23) Both integers are positive for the scales considered here. In particular,
q = 1 − 500 m D + O ( Z ∗ − 1 ) , q=1-\frac{500m}{D}+O(Z_*^{-1}), q = 1 − D 500 m + O ( Z ∗ − 1 ) , q ′ = 1000 m D + O ( Z ∗ − 1 ) , c m D ≤ q ′ ≤ C m D . (24) q'=\frac{1000m}{D}+O(Z_*^{-1}), \qquad c\frac{m}{D}\le q'\le C\frac{m}{D}. \tag*{(24)} q ′ = D 1000 m + O ( Z ∗ − 1 ) , c D m ≤ q ′ ≤ C D m . (24) The primary and secondary nuclei of site i i i X i X_i X i X i + ρ i e z X_i+\rho_i e_z X i + ρ i e z
r i = ∣ y − X i ∣ , V i ( y ) = q r i + q ′ ∣ y − X i − ρ i e z ∣ , r_i=\lvert y-X_i\rvert,\qquad V_i(y)=\frac{q}{r_i}+\frac{q'}{\lvert y-X_i-\rho_i e_z\rvert}, r i = ∣ y − X i ∣ , V i ( y ) = r i q + ∣ y − X i − ρ i e z ∣ q ′ , h = − 1 2 Δ − ∑ i = 1 m V i . (25) h=-\frac{1}{2}\Delta-\sum_{i=1}^{m}V_i. \tag*{(25)} h = − 2 1 Δ − i = 1 ∑ m V i . (25) These are the one-electron operators obtained from physical coordinates by y = Z ∗ x y=Z_*x y = Z ∗ x Z ∗ 2 Z_*^2 Z ∗ 2
Fix a nondecreasing C 2 C^2 C 2 χ : [ 0 , ∞ ) → [ 0 , 1 ] \chi:[0,\infty)\to[0,1] χ : [ 0 , ∞ ) → [ 0 , 1 ] [ 0 , 1 / 16 ] [0,1/16] [ 0 , 1/16 ] [ 1 / 8 , ∞ ) [1/8,\infty) [ 1/8 , ∞ ) ( t − 1 / 16 ) 2 ( 1 / 8 − t ) 2 (t-1/16)^2(1/8-t)^2 ( t − 1/16 ) 2 ( 1/8 − t ) 2
χ i ( y ) = χ ( r i / D ) , W i = ∑ k ≠ i χ k V k , h i = − 1 2 Δ − V i − W i . (26) \chi_i(y)=\chi(r_i/D),\qquad W_i=\sum_{k\ne i}\chi_kV_k,\qquad h_i=-\frac{1}{2}\Delta-V_i-W_i. \tag*{(26)} χ i ( y ) = χ ( r i / D ) , W i = k = i ∑ χ k V k , h i = − 2 1 Δ − V i − W i . (26) The caps remove the other nuclear singularities from h i h_i h i
We use the three-dimensional Sobolev bounds ∥ u ∥ 6 ≤ C ∥ u ∥ H 1 \|u\|_6 \le C\|u\|_{H^1} ∥ u ∥ 6 ≤ C ∥ u ∥ H 1 ∥ u ∥ ∞ ≤ C ∥ u ∥ H 2 \|u\|_\infty\le C\|u\|_{H^2} ∥ u ∥ ∞ ≤ C ∥ u ∥ H 2
∥ u ∣ ⋅ − a ∣ ∥ 2 ≤ 2 ∥ ∇ u ∥ 2 ( a ∈ R 3 ) . (27) \left\|\frac{u}{|\cdot-a|}\right\|_2 \le2\|\nabla u\|_2 \qquad(a \in\mathbb{R}^3). \tag*{(27)} ∣ ⋅ − a ∣ u 2 ≤ 2∥∇ u ∥ 2 ( a ∈ R 3 ) . (27) Indeed, integrate div ( ( y − a ) / ∣ y − a ∣ 2 ) = ∣ y − a ∣ − 2 \operatorname{div}((y-a)/|y-a|^2)=|y-a|^{-2} div (( y − a ) /∣ y − a ∣ 2 ) = ∣ y − a ∣ − 2 ∣ u ∣ 2 |u|^2 ∣ u ∣ 2 u u u
∥ u ∣ ⋅ − a ∣ ∥ 2 2 ≤ 2 ∥ u ∣ ⋅ − a ∣ ∥ 2 ∥ ∇ u ∥ 2 . \left\|\frac{u}{|\cdot-a|}\right\|_2^2 \le2\left\|\frac{u}{|\cdot-a|}\right\|_2\|\nabla u\|_2. ∣ ⋅ − a ∣ u 2 2 ≤ 2 ∣ ⋅ − a ∣ u 2 ∥∇ u ∥ 2 . Approximation proves (27) on H 1 H^1 H 1 H 1 H^1 H 1 L 2 L^2 L 2
Proposition 3.1 (Uniform isolated-well estimates). The operators h i h_i h i H 2 ( R 3 ) H^2(\mathbb{R}^3) H 2 ( R 3 ) H 1 ( R 3 ) H^1(\mathbb{R}^3) H 1 ( R 3 ) λ i \lambda_i λ i ψ i \psi_i ψ i g > 0 g>0 g > 0 λ i + g \lambda_i+g λ i + g ϕ i ( y ) = π − 1 / 2 e − r i \phi_i(y)=\pi^{-1/2}e^{-r_i} ϕ i ( y ) = π − 1/2 e − r i
∣ λ i + 1 / 2 ∣ ≤ C m D , ∥ ψ i − ϕ i ∥ H 2 ≤ C m D . (28) |\lambda_i+1/2| \le C\frac{m}{D}, \qquad\|\psi_i-\phi_i\|_{H^2} \le C\frac{m}{D}. \tag*{(28)} ∣ λ i + 1/2∣ ≤ C D m , ∥ ψ i − ϕ i ∥ H 2 ≤ C D m . (28) Moreover, for derivative orders j = 0 , 1 , 2 j=0,1,2 j = 0 , 1 , 2
∥ D j W i ∥ ∞ ≤ C m D j + 1 , 0 ≤ W i ≤ 100 m D . (29) \|D^jW_i\|_\infty\le C\frac{m}{D^{j+1}}, \qquad0\le W_i\le\frac{100m}{D}. \tag*{(29)} ∥ D j W i ∥ ∞ ≤ C D j + 1 m , 0 ≤ W i ≤ D 100 m . (29) These assertions hold before tuning and uniformly throughout [ 1 , 4 ] m [1,4]^m [ 1 , 4 ] m
Proof. On the nonzero support of χ k \chi_k χ k r k ≥ D / 16 r_k\ge D/16 r k ≥ D /16 ρ k ≤ 4 \rho_k\le4 ρ k ≤ 4
∣ y − X k − ρ k e z ∣ ≥ r k − 4 ≥ r k / 2 |y-X_k-\rho_k e_z|\ge r_k-4\ge r_k/2 ∣ y − X k − ρ k e z ∣ ≥ r k − 4 ≥ r k /2 for large D D D ∥ D j ( χ k V k ) ∥ ∞ ≤ C D − j − 1 \|D^j(\chi_kV_k)\|_\infty\le CD^{-j-1} ∥ D j ( χ k V k ) ∥ ∞ ≤ C D − j − 1 q ≤ 1 q\le1 q ≤ 1 q ′ ≤ 1 q'\le1 q ′ ≤ 1 D D D 100 / D 100/D 100/ D
Put H i 0 = − Δ / 2 − r i − 1 H_i^0=-\Delta/2-r_i^{-1} H i 0 = − Δ/2 − r i − 1 B i = h i − H i 0 B_i=h_i-H_i^0 B i = h i − H i 0 (29) give
∥ B i u ∥ 2 ≤ C m D ∥ u ∥ H 1 . (30) \|B_i u\|_2\le C\frac{m}{D}\|u\|_{H^1}. \tag*{(30)} ∥ B i u ∥ 2 ≤ C D m ∥ u ∥ H 1 . (30) For any potential in these single-well operators, Hardy and Fourier interpolation give, for each a > 0 a>0 a > 0 ∥ V u ∥ 2 ≤ a ∥ Δ u ∥ 2 + C a ∥ u ∥ 2 \|Vu\|_2\le a\|\Delta u\|_2+C_a\|u\|_2 ∥ V u ∥ 2 ≤ a ∥Δ u ∥ 2 + C a ∥ u ∥ 2 [15 ] therefore gives domain H 2 H^2 H 2
C − 1 ∥ u ∥ H 2 ≤ ∥ h i u ∥ 2 + C ∥ u ∥ 2 ≤ C ∥ u ∥ H 2 . C^{-1}\|u\|_{H^2}\le\|h_i u\|_2+C\|u\|_2\le C\|u\|_{H^2}. C − 1 ∥ u ∥ H 2 ≤ ∥ h i u ∥ 2 + C ∥ u ∥ 2 ≤ C ∥ u ∥ H 2 . The corresponding statement holds for H i 0 H_i^0 H i 0 ∥ u ∥ H 1 2 \|u\|_{H^1}^2 ∥ u ∥ H 1 2
The hydrogenic ground state and its isolation follow without an excited-state formula. Direct integration by parts gives
⟨ u , ( H i 0 + 1 / 2 ) u ⟩ = 1 2 ∥ ∇ u + y − X i r i u ∥ 2 2 . (31) \langle u,(H_i^0+1/2)u\rangle= \frac{1}{2}\left\|\nabla u+\frac{y-X_i}{r_i}u\right\|_2^2. \tag*{(31)} ⟨ u , ( H i 0 + 1/2 ) u ⟩ = 2 1 ∇ u + r i y − X i u 2 2 . (31) Its zero vectors are exactly the multiples of ϕ i \phi_i ϕ i ϕ i \phi_i ϕ i − 1 / 2 -1/2 − 1/2 H 1 H^1 H 1 a a a C a ∥ u ∥ H 1 2 Ca\|u\|_{H^1}^2 C a ∥ u ∥ H 1 2 R R R R − 1 ∥ u ∥ 2 2 R^{-1}\|u\|_2^2 R − 1 ∥ u ∥ 2 2
⟨ u , H i 0 u ⟩ ≤ − 1 / 2 , ⟨ u , H i 0 u ⟩ ≥ − 1 2 ∥ u ∥ 2 2 ≥ − 1 / 2. \langle u,H_i^0u\rangle\leq-1/2,\qquad\langle u,H_i^0u\rangle\geq-\frac{1}{2}\|u\|_2^2\geq-1/2. ⟨ u , H i 0 u ⟩ ≤ − 1/2 , ⟨ u , H i 0 u ⟩ ≥ − 2 1 ∥ u ∥ 2 2 ≥ − 1/2. Equality forces norm one and a multiple of ϕ i \phi_i ϕ i g 0 > 0 g_0>0 g 0 > 0
By (30) , after a fixed positive shift the perturbed form lies between 1 − C m / D 1-Cm/D 1 − C m / D 1 + C m / D 1+Cm/D 1 + C m / D C m / D Cm/D C m / D − 1 / 2 -1/2 − 1/2 − 1 / 2 + g 0 − C m / D -1/2+g_0-Cm/D − 1/2 + g 0 − C m / D W i W_i W i H 1 H^1 H 1 u u u
λ i ∥ u ∥ 2 2 ≤ ⟨ u , h i u ⟩ ≤ λ i < 0. \lambda_i\|u\|_2^2\leq\langle u,h_i u\rangle\leq\lambda_i<0. λ i ∥ u ∥ 2 2 ≤ ⟨ u , h i u ⟩ ≤ λ i < 0. Since ∥ u ∥ 2 ≤ 1 \|u\|_2\leq1 ∥ u ∥ 2 ≤ 1 ∥ u ∥ 2 = 1 \|u\|_2=1 ∥ u ∥ 2 = 1 m / D m/D m / D g > 0 g>0 g > 0
The form bounds give ∥ ψ i ∥ H 1 ≤ C \|\psi_i\|_{H^1}\leq C ∥ ψ i ∥ H 1 ≤ C
( H i 0 + 1 / 2 ) ψ i = ( λ i + 1 / 2 ) ψ i − B i ψ i , (H_i^0+1/2)\psi_i=(\lambda_i+1/2)\psi_i-B_i\psi_i, ( H i 0 + 1/2 ) ψ i = ( λ i + 1/2 ) ψ i − B i ψ i , with right side of norm at most C m / D Cm/D C m / D ϕ i ⊥ \phi_i^\perp ϕ i ⊥ H i 0 + 1 / 2 H_i^0+1/2 H i 0 + 1/2 L 2 L^2 L 2 H 2 H^2 H 2 L 2 L^2 L 2 (3.10) . The orthogonal component of ψ i \psi_i ψ i H 2 H^2 H 2 O ( m / D ) O(m/D) O ( m / D ) ϕ i \phi_i ϕ i ϕ i \phi_i ϕ i O ( ( m / D ) 2 ) O((m/D)^2) O (( m / D ) 2 ) (28) and the uniform sup bound. Positivity also holds at the poles by continuity and this sup-norm approximation, since both poles lie within distance four of X i X_i X i
Moving the secondary nucleus
The derivative of a moving Coulomb potential is not an L 2 L^2 L 2
Lemma 3.2 (Bracketing and uniform secant bounds). With centers and rounded charges fixed, for every choice of the other parameters, λ i ( ρ i = 1 ) < − 1 / 2 < λ i ( ρ i = 4 ) \lambda_i(\rho_i=1)<-1/2<\lambda_i(\rho_i=4) λ i ( ρ i = 1 ) < − 1/2 < λ i ( ρ i = 4 ) 1 ≤ a < b ≤ 4 1\leq a<b\leq4 1 ≤ a < b ≤ 4
c q ′ ( b − a ) ≤ λ i ( b ) − λ i ( a ) ≤ C q ′ ( b − a ) . (32) cq'(b-a)\leq\lambda_i(b)-\lambda_i(a)\leq Cq'(b-a). \tag*{(32)} c q ′ ( b − a ) ≤ λ i ( b ) − λ i ( a ) ≤ C q ′ ( b − a ) . (32) If instead ρ i \rho_i ρ i
∣ λ i ( ρ ) − λ i ( ρ ~ ) ∣ ≤ C q ′ m D 2 ∥ ρ − ρ ~ ∥ ∞ . (33) |\lambda_i(\rho)-\lambda_i(\widetilde{\rho})|\leq Cq'\frac{m}{D^2}\|\rho-\widetilde{\rho}\|_\infty. \tag*{(33)} ∣ λ i ( ρ ) − λ i ( ρ ) ∣ ≤ C q ′ D 2 m ∥ ρ − ρ ∥ ∞ . (33) Proof. Taking the inner product of the eigen-equation with ϕ i \phi_i ϕ i
λ i = − 1 2 + ( 1 − q ) − q ′ f ( ρ i ) − ⟨ ϕ i , W i ϕ i ⟩ + O ( ( m / D ) 2 ) . (34) \lambda_i=-\frac{1}{2}+(1-q)-q'f(\rho_i)-\langle\phi_i,W_i\phi_i\rangle+O((m/D)^2). \tag*{(34)} λ i = − 2 1 + ( 1 − q ) − q ′ f ( ρ i ) − ⟨ ϕ i , W i ϕ i ⟩ + O (( m / D ) 2 ) . (34) where
f ( ρ ) = ∫ R 3 π − 1 e − 2 ∣ y ∣ ∣ y − ρ e z ∣ d y . f(\rho)=\int_{\mathbb{R}^3}\frac{\pi^{-1}e^{-2|y|}}{|y-\rho e_z|}\,dy. f ( ρ ) = ∫ R 3 ∣ y − ρ e z ∣ π − 1 e − 2∣ y ∣ d y . Here ⟨ ϕ i , ψ i ⟩ = 1 + O ( ( m / D ) 2 ) \langle\phi_i,\psi_i\rangle=1+O((m/D)^2) ⟨ ϕ i , ψ i ⟩ = 1 + O (( m / D ) 2 ) ψ i \psi_i ψ i ϕ i \phi_i ϕ i O ( ( m / D ) 2 ) O((m/D)^2) O (( m / D ) 2 ) (30) and (28) . Also ⟨ ϕ i , r i − 1 ϕ i ⟩ = 1 \langle\phi_i,r_i^{-1}\phi_i\rangle=1 ⟨ ϕ i , r i − 1 ϕ i ⟩ = 1 ∣ r ω − ρ e z ∣ − 1 |r\omega-\rho e_z|^{-1} ∣ r ω − ρ e z ∣ − 1 1 / max ( r , ρ ) 1/\max(r,\rho) 1/ max ( r , ρ ) ω ⋅ e z \omega\cdot e_z ω ⋅ e z
f ( ρ ) = ρ − 1 ∫ 0 ρ 4 r 2 e − 2 r d r + ∫ ρ ∞ 4 r e − 2 r d r , (35) f(\rho)=\rho^{-1}\int_0^\rho4r^2e^{-2r}\,dr+\int_\rho^\infty4re^{-2r}\,dr, \tag*{(35)} f ( ρ ) = ρ − 1 ∫ 0 ρ 4 r 2 e − 2 r d r + ∫ ρ ∞ 4 r e − 2 r d r , (35) f ′ ( ρ ) = − ρ − 2 ∫ 0 ρ 4 r 2 e − 2 r d r . (36) f'(\rho)=-\rho^{-2}\int_0^\rho4r^2e^{-2r}\,dr. \tag*{(36)} f ′ ( ρ ) = − ρ − 2 ∫ 0 ρ 4 r 2 e − 2 r d r . (36) In particular,
f ( 1 ) = 1 − 2 e − 2 > 0.7 , f ( 4 ) ≤ 1 4 , 1 − 5 e − 2 16 ≤ − f ′ ( ρ ) ≤ 1 ( 1 ≤ ρ ≤ 4 ) . f(1)=1-2e^{-2}>0.7,\qquad f(4)\leq\frac{1}{4},\qquad\frac{1-5e^{-2}}{16}\leq-f'(\rho)\leq1\quad(1\leq\rho\leq4). f ( 1 ) = 1 − 2 e − 2 > 0.7 , f ( 4 ) ≤ 4 1 , 16 1 − 5 e − 2 ≤ − f ′ ( ρ ) ≤ 1 ( 1 ≤ ρ ≤ 4 ) . At the first endpoint the leading shift in (34) is at most ( 500 − 700 ) m / D (500-700)m/D ( 500 − 700 ) m / D ( 500 − 250 − 100 ) m / D (500-250-100)m/D ( 500 − 250 − 100 ) m / D
For the secant bound, fix a ground function u u u A ( t ) = X i + t e z A(t)=X_i+te_z A ( t ) = X i + t e z ∣ y − A ( t ) ∣ − 2 |y-A(t)|^{-2} ∣ y − A ( t ) ∣ − 2
∫ ∣ u 2 − ϕ i 2 ∣ ∣ y − A ( t ) ∣ 2 d y ≤ ∥ u − ϕ i ⋅ − A ( t ) ∥ 2 ∥ u + ϕ i ⋅ − A ( t ) ∥ 2 ≤ 4 ∥ ∇ ( u − ϕ i ) ∥ 2 ( ∥ ∇ u ∥ 2 + ∥ ∇ ϕ i ∥ 2 ) ≤ C m D . (37) \int\frac{|u^2-\phi_i^2|}{|y-A(t)|^2}\,dy\leq\left\|\frac{u-\phi_i}{\cdot-A(t)}\right\|_2\left\|\frac{u+\phi_i}{\cdot-A(t)}\right\|_2
\leq4\|\nabla(u-\phi_i)\|_2(\|\nabla u\|_2+\|\nabla\phi_i\|_2)\leq C\frac{m}{D}. \tag*{(37)} ∫ ∣ y − A ( t ) ∣ 2 ∣ u 2 − ϕ i 2 ∣ d y ≤ ⋅ − A ( t ) u − ϕ i 2 ⋅ − A ( t ) u + ϕ i 2 ≤ 4∥∇ ( u − ϕ i ) ∥ 2 ( ∥∇ u ∥ 2 + ∥∇ ϕ i ∥ 2 ) ≤ C D m . (37) The same bound with u 2 u^2 u 2 y y y t t t ϕ i 2 \phi_i^2 ϕ i 2 C q ′ ( m / D ) ( b − a ) Cq'(m/D)(b-a) C q ′ ( m / D ) ( b − a )
⟨ ψ i ( b ) , ( h i ( b ) − h i ( a ) ) ψ i ( b ) ⟩ ≤ λ i ( b ) − λ i ( a ) ≤ ⟨ ψ i ( a ) , ( h i ( b ) − h i ( a ) ) ψ i ( a ) ⟩ . \langle\psi_i(b),(h_i(b)-h_i(a))\psi_i(b)\rangle\leq\lambda_i(b)-\lambda_i(a)\leq\langle\psi_i(a),(h_i(b)-h_i(a))\psi_i(a)\rangle. ⟨ ψ i ( b ) , ( h i ( b ) − h i ( a )) ψ i ( b )⟩ ≤ λ i ( b ) − λ i ( a ) ≤ ⟨ ψ i ( a ) , ( h i ( b ) − h i ( a )) ψ i ( a )⟩ . Both bounding quantities differ from − q ′ ( f ( b ) − f ( a ) ) -q'(f(b)-f(a)) − q ′ ( f ( b ) − f ( a )) (32) .
For k ≠ i k\ne i k = i χ k V k \chi_kV_k χ k V k D / 32 D/32 D /32 C q ′ D − 2 ∣ ρ k − ρ ~ k ∣ Cq'D^{-2}|\rho_k-\tilde{\rho}_k| C q ′ D − 2 ∣ ρ k − ρ ~ k ∣ (33) .
Lemma 3.3 (Crude tails before tuning). Uniformly for all ρ ∈ [ 1 , 4 ] m \rho\in[1,4]^m ρ ∈ [ 1 , 4 ] m
∣ ψ i ( y ) ∣ ≤ C exp [ − .99 ( r i − .01 D ) ] . (38) |\psi_i(y)| \le C\exp[-.99(r_i-.01D)]. \tag*{(38)} ∣ ψ i ( y ) ∣ ≤ C exp [ − .99 ( r i − .01 D )] . (38) and, for sufficiently large D D D
∥ ψ i ∥ H 1 ( { r i > .65 D } ) ≤ e − .61 D . (39) \|\psi_i\|_{H^1(\{r_i>.65D\})} \le e^{-.61D}. \tag*{(39)} ∥ ψ i ∥ H 1 ({ r i > .65 D }) ≤ e − .61 D . (39) Proof. On r i ≥ .01 D r_i \ge.01D r i ≥ .01 D V i + W i = O ( m / D ) V_i+W_i=O(m/D) V i + W i = O ( m / D ) a = .99 a=.99 a = .99 b ( y ) = C exp [ − a ( r i − .01 D ) ] b(y)=C\exp[-a(r_i-.01D)] b ( y ) = C exp [ − a ( r i − .01 D )]
( h i − λ i ) b b = − a 2 2 + a r i − V i − W i − λ i . \frac{(h_i-\lambda_i)b}{b}=-\frac{a^2}{2}+\frac{a}{r_i}-V_i-W_i-\lambda_i. b ( h i − λ i ) b = − 2 a 2 + r i a − V i − W i − λ i . Its constant part is ( 1 − a 2 ) / 2 + O ( m / D ) > 0 (1-a^2)/2+O(m/D)>0 ( 1 − a 2 ) /2 + O ( m / D ) > 0 h i − λ i h_i-\lambda_i h i − λ i C C C b ≥ ∣ ψ i ∣ b\ge|\psi_i| b ≥ ∣ ψ i ∣ H 2 ( R 3 ) H^2(\mathbb{R}^3) H 2 ( R 3 ) ε > 0 \varepsilon>0 ε > 0 b b b ψ i \psi_i ψ i b + ε ∓ ψ i b+\varepsilon\mp\psi_i b + ε ∓ ψ i ε \varepsilon ε .01 D .01D .01 D (38) .
For the gradient estimate choose a smooth cutoff ζ \zeta ζ r i ≤ .645 D r_i\le.645D r i ≤ .645 D r i ≥ .65 D r_i\ge.65D r i ≥ .65 D O ( D − 1 ) O(D^{-1}) O ( D − 1 ) ζ 2 ψ i \zeta^2\psi_i ζ 2 ψ i
1 2 ∥ ∇ ( ζ ψ i ) ∥ 2 2 = ∫ ( V i + W i + λ i ) ∣ ζ ψ i ∣ 2 d y + 1 2 ∫ ∣ ∇ ζ ∣ 2 ∣ ψ i ∣ 2 d y . \frac{1}{2}\|\nabla(\zeta\psi_i)\|_2^2=\int(V_i+W_i+\lambda_i)|\zeta\psi_i|^2\,dy+\frac{1}{2}\int|\nabla\zeta|^2|\psi_i|^2\,dy. 2 1 ∥∇ ( ζ ψ i ) ∥ 2 2 = ∫ ( V i + W i + λ i ) ∣ ζ ψ i ∣ 2 d y + 2 1 ∫ ∣∇ ζ ∣ 2 ∣ ψ i ∣ 2 d y . All coefficients on this support are bounded. Thus (38) bounds the exterior H 1 H^1 H 1 D D D exp [ − .99 ( .645 − .01 ) D ] \exp[-.99(.645-.01)D] exp [ − .99 ( .645 − .01 ) D ] .99 ( .645 − .01 ) = .62865 > .61 .99(.645-.01)=.62865>.61 .99 ( .645 − .01 ) = .62865 > .61 D D D
We give the local regularity argument used in the numerical procedure. It explains why the shrinking length scales near a moving Coulomb pole do not increase the required polynomial degree beyond O ( D ) O(D) O ( D )
Lemma 3.4 (Uniform analytic patches). Suppose u u u R 3 \mathbb{R}^3 R 3 and satisfies Δ u = F u \Delta u=Fu Δ u = F u there, where
∥ ∂ β F ∥ ∞ ≤ A ∣ β ∣ + 1 ∣ β ∣ ! for every multi-index β . (40) \|\partial^\beta F\|_\infty\le A^{|\beta|+1}|\beta|!\quad\text{for every multi-index }\beta. \tag*{(40)} ∥ ∂ β F ∥ ∞ ≤ A ∣ β ∣ + 1 ∣ β ∣ ! for every multi-index β . (40) There is a constant A 1 A_1 A 1 depending only on A A A such that
max ∣ α ∣ = k ∥ ∂ α u ∥ L ∞ ( B 1 / 4 ) ≤ ∥ u ∥ L 2 ( B 1 ) A 1 k + 1 k ! . (41) \max_{|\alpha|=k}\|\partial^\alpha u\|_{L^\infty(B_{1/4})}\le\|u\|_{L^2(B_1)}A_1^{k+1}k!. \tag*{(41)} ∣ α ∣ = k max ∥ ∂ α u ∥ L ∞ ( B 1/4 ) ≤ ∥ u ∥ L 2 ( B 1 ) A 1 k + 1 k ! . (41) On a sufficiently small fixed ball about the origin, Taylor polynomials consequently have geometrically decreasing errors in both value and first derivative, with constants depending only on A A A
Proof. For two concentric balls of radii between 1 / 2 1/2 1/2 1 1 1 h h h
∥ D 2 v ∥ 2 , i n n e r ≤ C ( ∥ Δ v ∥ 2 , o u t e r + h − 2 ∥ v ∥ 2 , o u t e r ) . (42) \lVert D^2v\rVert_{2,\mathrm{inner}} \le C\left(\lVert\Delta v\rVert_{2,\mathrm{outer}}+h^{-2}\lVert v\rVert_{2,\mathrm{outer}}\right). \tag*{(42)} ∥ D 2 v ∥ 2 , inner ≤ C ( ∥ Δ v ∥ 2 , outer + h − 2 ∥ v ∥ 2 , outer ) . (42) ∥ ∇ v ∥ 2 , i n n e r ≤ C ( h ∥ Δ v ∥ 2 , o u t e r + h − 1 ∥ v ∥ 2 , o u t e r ) . (43) \lVert\nabla v\rVert_{2,\mathrm{inner}} \le C\left(h\lVert\Delta v\rVert_{2,\mathrm{outer}}+h^{-1}\lVert v\rVert_{2,\mathrm{outer}}\right). \tag*{(43)} ∥ ∇ v ∥ 2 , inner ≤ C ( h ∥ Δ v ∥ 2 , outer + h − 1 ∥ v ∥ 2 , outer ) . (43) To obtain them, insert smooth cutoffs with first and second derivatives O ( h − 1 ) O(h^{-1}) O ( h − 1 ) O ( h − 2 ) O(h^{-2}) O ( h − 2 ) a b ≤ ( h a ) 2 / 2 + ( b / h ) 2 / 2 ab\le(ha)^2/2+(b/h)^2/2 ab ≤ ( ha ) 2 /2 + ( b / h ) 2 /2 (43) . Apply the whole-space Fourier inequality ∥ D 2 w ∥ 2 ≤ C ∥ Δ w ∥ 2 \lVert D^2w\rVert_2\le C\lVert\Delta w\rVert_2 ∥ D 2 w ∥ 2 ≤ C ∥ Δ w ∥ 2 (42) . Approximation proves these estimates for weak solutions. Successive localized Laplacian regularity makes u u u F F F
Fix an integer ℓ ≥ 2 \ell\ge2 ℓ ≥ 2 r j = 1 − j / ( 2 ℓ ) r_j=1-j/(2\ell) r j = 1 − j / ( 2 ℓ )
T j = max ∣ α ∣ = j ∥ ∂ α u ∥ L 2 ( B r j ) , M = ∥ u ∥ L 2 ( B 1 ) . T_j=\max_{|\alpha|=j}\lVert\partial^\alpha u\rVert_{L^2(B_{r_j})},\qquad M=\lVert u\rVert_{L^2(B_1)}. T j = ∣ α ∣ = j max ∥ ∂ α u ∥ L 2 ( B r j ) , M = ∥ u ∥ L 2 ( B 1 ) . The first-derivative bound gives T 1 ≤ C ( ℓ + A / ℓ ) M T_1\le C(\ell+A/\ell)M T 1 ≤ C ( ℓ + A / ℓ ) M j ≤ ℓ − 2 j\le\ell-2 j ≤ ℓ − 2 (42) to an order-j j j B r j B_{r_j} B r j B r j + 2 B_{r_{j+2}} B r j + 2
T j + 2 ≤ C ( ∑ b = 0 j ( j b ) A b + 1 b ! T j − b + ℓ 2 T j ) . (44) T_{j+2}\le C\left(\sum_{b=0}^{j}\binom{j}{b}A^{b+1}b!T_{j-b}+\ell^2T_j\right). \tag*{(44)} T j + 2 ≤ C ( b = 0 ∑ j ( b j ) A b + 1 b ! T j − b + ℓ 2 T j ) . (44) Lower-order derivatives are available on the larger balls B r j − b B_{r_{j-b}} B r j − b b b b ( j b ) \binom{j}{b} ( b j ) T s ≤ M ( K ℓ ) s T_s\le M(K\ell)^s T s ≤ M ( K ℓ ) s
A M ( K ℓ ) j ∑ b = 0 j ( A / K ) b ≤ 2 A M ( K ℓ ) j AM(K\ell)^j\sum_{b=0}^{j}(A/K)^b\le2AM(K\ell)^j A M ( K ℓ ) j b = 0 ∑ j ( A / K ) b ≤ 2 A M ( K ℓ ) j for K ≥ 2 A K\ge2A K ≥ 2 A ( j b ) b ! ≤ ℓ b \binom{j}{b}b!\le\ell^b ( b j ) b ! ≤ ℓ b K K K C ℓ 2 C\ell^2 C ℓ 2 ( K ℓ ) 2 (K\ell)^2 ( K ℓ ) 2 T j ≤ M ( K ℓ ) j T_j\le M(K\ell)^j T j ≤ M ( K ℓ ) j
Set ℓ = k + 2 \ell=k+2 ℓ = k + 2 k k k B 1 / 4 B_{1/4} B 1/4 C M [ K ( k + 2 ) ] k + 2 CM[K(k+2)]^{k+2} C M [ K ( k + 2 ) ] k + 2 k ! ≥ ( k / e ) k k!\ge(k/e)^k k ! ≥ ( k / e ) k (41) . Taylor’s formula, summed over multi-indices, bounds the degree-L L L ∣ z ∣ ≤ a |z|\le a ∣ z ∣ ≤ a ( C 1 a ) L + 1 (C_1a)^{L+1} ( C 1 a ) L + 1 L L L a a a C 1 a < 1 C_1a<1 C 1 a < 1
A polynomial-size graded Ritz space
Graded meshes with increasing polynomial degree are a standard way to combine singular local behavior with exponential approximation away from the singularities. For related eigenvalue approximation results with singular potentials, see Maday and Marcati [6 ] . Their bounded-domain discontinuous-Galerkin estimates are not an oracle for the present moving attractive Coulomb wells. We construct the needed conforming space and then certify its matrix arithmetic separately. More specifically, the products of tensor-linear hats with local polynomials below form a partition-of-unity approximation space, as in Melenk and Babuška [7 ] . We give the cutoff and derivative estimates explicitly because the tensor cells can have arbitrarily large aspect ratios.
This is a form-domain approximation: the piecewise polynomial functions need only belong to H 1 H^1 H 1
Lemma 3.5 (Exponentially accurate Ritz approximation). For each i i i D D D V i ⊂ H 1 ( R 3 ) V_i \subset H^1(\mathbb{R}^3) V i ⊂ H 1 ( R 3 ) D D D r i < .8 D r_i < .8D r i < .8 D
inf v ∈ V i ∥ v − ψ i ∥ H 1 ≤ e − .58 D . (45) \inf_{v\in V_i} \lVert v-\psi_i\rVert_{H^1} \le e^{-.58D}. \tag*{(45)} v ∈ V i inf ∥ v − ψ i ∥ H 1 ≤ e − .58 D . (45) Its lowest Ritz value λ i R \lambda_i^{\mathrm{R}} λ i R satisfies
0 ≤ λ i R − λ i ≤ C e − 1.16 D . (46) 0 \le\lambda_i^{\mathrm{R}}-\lambda_i \le Ce^{-1.16D}. \tag*{(46)} 0 ≤ λ i R − λ i ≤ C e − 1.16 D . (46) The construction and constants are uniform over rational parameters throughout [ 1 , 4 ] m [1,4]^m [ 1 , 4 ] m
Proof. Work in local coordinates z = y − X i z=y-X_i z = y − X i [ − D , D ] 3 [-D,D]^3 [ − D , D ] 3 0 0 0 ρ i e z \rho_i e_z ρ i e z δ = 2 − 8 D \delta=2^{-8D} δ = 2 − 8 D ξ > 0 \xi>0 ξ > 0 B B B [ − D , D ] [-D,D] [ − D , D ] I I I
∣ I ∣ ≤ ξ min { 1 , dist ( I , B ) + δ } . (47) \lvert I\rvert\le\xi\min\{1,\operatorname{dist}(I,B)+\delta\}. \tag*{(47)} ∣ I ∣ ≤ ξ min { 1 , dist ( I , B ) + δ } . (47) Every leaf length is at least ξ δ / 2 \xi\delta/2 ξ δ /2 O ( D ) O(D) O ( D ) h h h h / ξ h/\xi h / ξ O ( log ( D / δ ) ) = O ( D + log D ) O(\log(D/\delta))=O(D+\log D) O ( log ( D / δ )) = O ( D + log D ) O ( D ) O(D) O ( D ) O ( D 3 ) O(D^3) O ( D 3 )
Let b w b_w b w w w w
∣ w ∣ < .73 D , ∣ w ∣ > δ , ∣ w − ρ i e z ∣ > δ , (48) \lvert w\rvert<.73D,\qquad\lvert w\rvert>\delta,\qquad\lvert w-\rho_i e_z\rvert>\delta, \tag*{(48)} ∣ w ∣ < .73 D , ∣ w ∣ > δ , ∣ w − ρ i e z ∣ > δ , (48) and take b w ( z ) ( z − w ) α b_w(z)(z-w)^\alpha b w ( z ) ( z − w ) α ∣ α ∣ ≤ L 0 \lvert\alpha\rvert\le L_0 ∣ α ∣ ≤ L 0 L 0 L_0 L 0 D D D V i V_i V i O ( D 6 ) O(D^6) O ( D 6 )
Here is the support estimate that avoids any aspect-ratio assumption. For either pole a a a w w w
∣ I ∣ ≤ ξ min { 1 , ∣ w j − a j ∣ + δ } . \lvert I\rvert\le\xi\min\{1,\lvert w_j-a_j\rvert+\delta\}. ∣ I ∣ ≤ ξ min { 1 , ∣ w j − a j ∣ + δ } . The support radius R w h a t R_w^{\mathrm{hat}} R w hat
R w h a t ≤ ξ ( ∣ w − a ∣ + 3 δ ) ≤ ( 1 + 3 ) ξ ∣ w − a ∣ , R_w^{\mathrm{hat}}\le\xi(\lvert w-a\rvert+\sqrt{3}\delta)\le(1+\sqrt{3})\xi\lvert w-a\rvert, R w hat ≤ ξ (∣ w − a ∣ + 3 δ ) ≤ ( 1 + 3 ) ξ ∣ w − a ∣ , R w h a t ≤ 3 ξ . (49) R_w^{\mathrm{hat}}\le\sqrt{3}\xi. \tag*{(49)} R w hat ≤ 3 ξ . (49) For d w = min { 1 , ∣ w ∣ , ∣ w − ρ i e z ∣ } d_w=\min\{1,\lvert w\rvert,\lvert w-\rho_i e_z\rvert\} d w = min { 1 , ∣ w ∣ , ∣ w − ρ i e z ∣} R w h a t ≤ C ξ d w R_w^{\mathrm{hat}}\le C\xi d_w R w hat ≤ C ξ d w D D D ∣ z ∣ < .8 D \lvert z\rvert<.8D ∣ z ∣ < .8 D
Choose a fixed c < 1 / 10 c<1/10 c < 1/10 R = c d w R=cd_w R = c d w B ( w , R ) B(w,R) B ( w , R ) D − 2 S − .73 D − c > D / 8 D-2S-.73D-c>D/8 D − 2 S − .73 D − c > D /8 R R R Δ u = F u \Delta u=Fu Δ u = F u (40) with an absolute A A A R / dist ( w , a ) ≤ c R/\operatorname{dist}(w,a) \le c R / dist ( w , a ) ≤ c R 2 R^2 R 2 R 2 / dist ( w , a ) R^2/\operatorname{dist}(w,a) R 2 / dist ( w , a ) C m / D Cm/D C m / D u u u
Choose ξ \xi ξ c c c L 0 L_0 L 0 D D D p w p_w p w
∣ ψ i − p w ∣ + R ∣ ∇ ψ i − ∇ p w ∣ ≤ e − 20 D on supp b w . (50) |\psi_i-p_w|+R|\nabla\psi_i-\nabla p_w|\le e^{-20D}\quad\text{on supp }b_w. \tag*{(50)} ∣ ψ i − p w ∣ + R ∣∇ ψ i − ∇ p w ∣ ≤ e − 20 D on supp b w . (50) Increasing the fixed multiple absorbs the polynomial factor in the differentiated remainder. The Taylor coefficients establish existence of an approximant; the algorithm will not compute them.
To remove tiny neighborhoods of the poles and the exterior, let a o u t a_{\mathrm{out}} a out .66 D .66D .66 D .72 D .72D .72 D a i n a_{\mathrm{in}} a in δ \delta δ 2 δ 2\delta 2 δ
a o u t ( ∣ z ∣ ) a i n ( ∣ z ∣ ) a i n ( ∣ z − ρ i e z ∣ ) a_{\mathrm{out}}(|z|)\,a_{\mathrm{in}}(|z|)\,a_{\mathrm{in}}(|z-\rho_i e_z|) a out ( ∣ z ∣ ) a in ( ∣ z ∣ ) a in ( ∣ z − ρ i e z ∣ ) at all grid nodes to get weights c w ∈ [ 0 , 1 ] c_w\in[0,1] c w ∈ [ 0 , 1 ] b = ∑ w c w b w b=\sum_w c_wb_w b = ∑ w c w b w b b b O ( δ ) O(\delta) O ( δ ) ∣ z ∣ > .65 D |z|>.65D ∣ z ∣ > .65 D 2 δ 2\delta 2 δ O ( ξ δ ) O(\xi\delta) O ( ξ δ ) ξ \xi ξ
Each first derivative of a tensor-linear interpolant is a convex combination of edge secants in that coordinate. Consequently ∣ ∇ b ∣ ≤ C / δ |\nabla b|\le C/\delta ∣∇ b ∣ ≤ C / δ ∣ ∇ b ∣ ≤ C / D |\nabla b|\le C/D ∣∇ b ∣ ≤ C / D ψ i \psi_i ψ i C δ 1 / 2 C\delta^{1/2} C δ 1/2 H 2 H^2 H 2
∥ ∇ ψ i ∥ L 2 ( B C δ ( a ) ) ≤ C δ ∥ ∇ ψ i ∥ 6 ≤ C δ . \|\nabla\psi_i\|_{L^2(B_{C\delta}(a))}\le C\delta\|\nabla\psi_i\|_6\le C\delta. ∥∇ ψ i ∥ L 2 ( B C δ ( a )) ≤ C δ ∥∇ ψ i ∥ 6 ≤ C δ . The exterior cost follows from (39) . Thus
∥ ψ i − b ψ i ∥ H 1 ≤ C ( δ + e − .61 D ) . (51) \|\psi_i-b\psi_i\|_{H^1}\le C\left(\sqrt{\delta}+e^{-.61D}\right). \tag*{(51)} ∥ ψ i − b ψ i ∥ H 1 ≤ C ( δ + e − .61 D ) . (51) The cutoff weights, like the Taylor coefficients, are only existence coefficients in the explicitly generated space.
Set v = ∑ w c w b w p w ∈ V i v=\sum_w c_wb_wp_w\in V_i v = ∑ w c w b w p w ∈ V i C / δ C/\delta C / δ R − 1 ≤ C / δ R^{-1}\le C/\delta R − 1 ≤ C / δ O ( D 3 ) O(D^3) O ( D 3 ) (50) gives
∥ b ψ i − v ∥ H 1 ≤ C D 3 / 2 δ − 1 e − 20 D . \|b\psi_i-v\|_{H^1}\le CD^{3/2}\delta^{-1}e^{-20D}. ∥ b ψ i − v ∥ H 1 ≤ C D 3/2 δ − 1 e − 20 D . Since δ = e − 4 ( log 2 ) D \sqrt{\delta}=e^{-4(\log2)D} δ = e − 4 ( l o g 2 ) D δ − 1 e − 20 D = e − ( 20 − 8 log 2 ) D \delta^{-1}e^{-20D}=e^{-(20-8\log2)D} δ − 1 e − 20 D = e − ( 20 − 8 l o g 2 ) D (51) prove (45) for large D D D
Finally, the form of h i − λ i h_i-\lambda_i h i − λ i H 1 H^1 H 1 e = v − ψ i e=v-\psi_i e = v − ψ i
⟨ v , h i v ⟩ ∥ v ∥ 2 2 − λ i = ⟨ e , ( h i − λ i ) e ⟩ ∥ v ∥ 2 2 ≤ C ∥ e ∥ H 1 2 . \frac{\langle v,h_iv\rangle}{\|v\|_2^2}-\lambda_i
=\frac{\langle e,(h_i-\lambda_i)e\rangle}{\|v\|_2^2}
\le C\|e\|_{H^1}^2. ∥ v ∥ 2 2 ⟨ v , h i v ⟩ − λ i = ∥ v ∥ 2 2 ⟨ e , ( h i − λ i ) e ⟩ ≤ C ∥ e ∥ H 1 2 . Here ∥ v ∥ 2 \|v\|_2 ∥ v ∥ 2 (46) .
Certified polynomial-bit eigenvalue evaluation
A small Ritz space by itself does not establish bit complexity. We next bound the conditioning loss and give certified matrix entries and eigenvalue enclosures. All arithmetic below is classical.
Proposition 3.6 (Uniform single-well eigenvalue oracle). Suppose the centers and the parameters in [ 1 , 4 ] m [1,4]^m [ 1 , 4 ] m q , q ′ q,q' q , q ′ B B B ℓ i ≤ λ i ≤ u i \ell_i \le\lambda_i \le u_i ℓ i ≤ λ i ≤ u i
u i − ℓ i ≤ ε D , ε D = 2 − ⌈ 8 D / 5 ⌉ < e − 1.1 D . (52) u_i-\ell_i \le\varepsilon_D,\qquad\varepsilon_D=2^{-\lceil8D/5\rceil}<e^{-1.1D}. \tag*{(52)} u i − ℓ i ≤ ε D , ε D = 2 − ⌈ 8 D /5 ⌉ < e − 1.1 D . (52) Its bit complexity is polynomial in D , m , B D,m,B D , m , B [ 1 , 4 ] m [1,4]^m [ 1 , 4 ] m
Proof. We divide the construction into exact polynomial arithmetic, certified Coulomb integration, and rational eigenvalue enclosure.
Exact matrices and a mass lower bound. Construct the basis in Theorem 3.5. All grid decisions are rational comparisons: one-dimensional distances in (47) are rational, and node membership in (48) is checked by comparing squared distances. Grid depth is O ( D + log D ) O(D+\log D) O ( D + log D ) O ( D ) O(D) O ( D ) b b b L L L O ( L b ) O(Lb) O ( L b ) b = O ( B + D + log D ) b=O(B+D+\log D) b = O ( B + D + log D ) L = O ( D ) L=O(D) L = O ( D )
Let f 1 , … , f r 0 f_1,\ldots,f_{r_0} f 1 , … , f r 0 r 0 = O ( D 6 ) r_0=O(D^6) r 0 = O ( D 6 )
M a b = ∫ f a f b d y , T a b = 1 2 ∫ ∇ f a ⋅ ∇ f b d y (53) M_{ab}=\int f_a f_b\,\mathrm{d}y,\qquad T_{ab}=\frac{1}{2}\int\nabla f_a\cdot\nabla f_b\,\mathrm{d}y \tag*{(53)} M ab = ∫ f a f b d y , T ab = 2 1 ∫ ∇ f a ⋅ ∇ f b d y (53) are exact rationals obtained by box integration. A product of two hats has only a fixed number of box pieces; alternatively, summing over the entire polynomial-size grid also suffices. Remove redundant functions by exact rational Gram-matrix elimination, retaining an independent subset. This gives a positive definite matrix M M M r ≤ r 0 r\le r_0 r ≤ r 0 T T T
Let B 0 B_0 B 0 M M M D , m , B D,m,B D , m , B Q Q Q Q ≤ 2 r 2 B 0 Q\le2^{r^2B_0} Q ≤ 2 r 2 B 0 Q M QM QM det ( Q M ) ≥ 1 \det(QM)\ge1 det ( QM ) ≥ 1 λ max ( M ) ≤ r 2 2 B 0 \lambda_{\max}(M)\le r2^{2B_0} λ m a x ( M ) ≤ r 2 2 B 0
μ = 2 − K M , K M = r 3 B 0 + ( r − 1 ) ( B 0 + ⌈ log 2 r ⌉ ) (54) \mu=2^{-K_M},\qquad K_M=r^3B_0+(r-1)\left(B_0+\lceil\log_2 r\rceil\right) \tag*{(54)} μ = 2 − K M , K M = r 3 B 0 + ( r − 1 ) ( B 0 + ⌈ log 2 r ⌉ ) (54) satisfies λ min ( M ) ≥ μ \lambda_{\min}(M)\ge\mu λ m i n ( M ) ≥ μ K M K_M K M
Certified Coulomb entries. The preceding determinant bound shows how many extra bits compensate for a poorly conditioned basis. It remains to compute each potential entry to that absolute accuracy. The graded supports avoid every pole, which permits a uniformly convergent kernel expansion on each box. On the supports of the basis functions every other cap equals one, so every potential entry is a finite sum of pure Coulomb integrals against polynomials on rational boxes. For each such box Q 0 Q_0 Q 0 a a a c 0 c_0 c 0
s = ∣ c 0 − a ∣ 2 , z ( y ) = 2 ( c 0 − a ) ⋅ ( y − c 0 ) + ∣ y − c 0 ∣ 2 s . s = \lvert c_0-a\rvert^2,\qquad z(y) = \frac{2(c_0-a)\cdot(y-c_0)+\lvert y-c_0\rvert^2}{s}. s = ∣ c 0 − a ∣ 2 , z ( y ) = s 2 ( c 0 − a ) ⋅ ( y − c 0 ) + ∣ y − c 0 ∣ 2 . Every contributing box is in the support of a retained hat. By (49) , its diameter is a fixed small fraction of its distance from either own pole. Remote poles are farther still. Decreasing the fixed ξ \xi ξ ∣ z ( y ) ∣ ≤ 1 / 2 \lvert z(y)\rvert\le1/2 ∣ z ( y )∣ ≤ 1/2
1 ∣ y − a ∣ = s − 1 / 2 ∑ j = 0 J ( − 1 / 2 j ) z ( y ) j + R J ( y ) , ∣ R J ( y ) ∣ ≤ s − 1 / 2 2 − J . (55) \frac{1}{\lvert y-a\rvert} = s^{-1/2}\sum_{j=0}^{J}\binom{-1/2}{j}z(y)^j+R_J(y),\qquad\lvert R_J(y)\rvert\le s^{-1/2}2^{-J}. \tag*{(55)} ∣ y − a ∣ 1 = s − 1/2 j = 0 ∑ J ( j − 1/2 ) z ( y ) j + R J ( y ) , ∣ R J ( y )∣ ≤ s − 1/2 2 − J . (55) The coefficient bound ∣ ( − 1 / 2 j ) ∣ = 4 − j ( 2 j j ) ≤ 1 \left\lvert\binom{-1/2}{j}\right\rvert=4^{-j}\binom{2j}{j}\le1 ( j − 1/2 ) = 4 − j ( j 2 j ) ≤ 1
Here s s s z z z log 2 + ( s − 1 / 2 ) \log_2^+(s^{-1/2}) log 2 + ( s − 1/2 ) D D D m m m B B B δ \delta δ s − 1 / 2 s^{-1/2} s − 1/2
We spell out an absolute-error certificate, so cancellation among polynomial terms causes no difficulty. If P P P Q 0 Q_0 Q 0
A Q 0 ≥ ∫ Q 0 ∣ P ( y ) ∣ d y . A_{Q_0}\ge\int_{Q_0}\lvert P(y)\rvert\,dy. A Q 0 ≥ ∫ Q 0 ∣ P ( y )∣ d y . One may take the volume times the sum of absolute coefficients, each multiplied by the appropriate coordinate bound to its degree. Its bit length is polynomial. Include the charge in this bound. If s − 1 / 2 ≤ 2 b s^{-1/2}\le2^b s − 1/2 ≤ 2 b A Q 0 2 b − J A_{Q_0}2^{b-J} A Q 0 2 b − J s − 1 / 2 s^{-1/2} s − 1/2 2 − q 0 2^{-q_0} 2 − q 0 2 A Q 0 2 − q 0 2A_{Q_0}2^{-q_0} 2 A Q 0 2 − q 0 (55) is at most two. For T 0 T_0 T 0
J ≥ p + b + ⌈ log 2 + A Q 0 ⌉ + ⌈ log 2 T 0 ⌉ + 3 , J\ge p+b+\left\lceil\log_2^+ A_{Q_0}\right\rceil+\left\lceil\log_2 T_0\right\rceil+3, J ≥ p + b + ⌈ log 2 + A Q 0 ⌉ + ⌈ log 2 T 0 ⌉ + 3 , q 0 ≥ p + ⌈ log 2 + A Q 0 ⌉ + ⌈ log 2 T 0 ⌉ + 4 q_0\ge p+\left\lceil\log_2^+ A_{Q_0}\right\rceil+\left\lceil\log_2 T_0\right\rceil+4 q 0 ≥ p + ⌈ log 2 + A Q 0 ⌉ + ⌈ log 2 T 0 ⌉ + 4 makes their total error smaller than 2 − p 2^{-p} 2 − p log 2 + t = max { 0 , log 2 t } \log_2^+t=\max\{0,\log_2t\} log 2 + t = max { 0 , log 2 t }
The polynomial z ( y ) j z(y)^j z ( y ) j 2 j 2j 2 j O ( ( D + J + 1 ) 3 ) O((D+J+1)^3) O (( D + J + 1 ) 3 ) D D D m m m B B B J J J q 0 q_0 q 0 p p p
Matrix error and exact eigenvalue decisions. Let A A A h i h_i h i A ~ \widetilde A A
τ = ε D μ 16 . \tau=\frac{\varepsilon_D\mu}{16}. τ = 16 ε D μ . The kinetic matrix is exact. Since a symmetric matrix’s operator norm is at most τ \tau τ
∥ M − 1 / 2 ( A ~ − A ) M − 1 / 2 ∥ ≤ ε D / 16. (56) \left\lVert M^{-1/2}(\widetilde A-A)M^{-1/2}\right\rVert\le\varepsilon_D/16. \tag*{(56)} M − 1/2 ( A − A ) M − 1/2 ≤ ε D /16. (56) The required entry precision has O ( D + K M + log r ) O(D + K_M + \log r) O ( D + K M + log r ) λ ~ \widetilde{\lambda} λ ( A ~ , M ) (\widetilde{A}, M) ( A , M ) (56) give
∣ λ ~ − λ i R ∣ ≤ ε D / 16. \left|\widetilde{\lambda}-\lambda_i^{R}\right| \le\varepsilon_D/16. λ − λ i R ≤ ε D /16. An initial rational bracket is [ − R 0 , R 0 ] [-R_0,R_0] [ − R 0 , R 0 ]
R 0 = 1 + r max a , b ∣ A ~ a b ∣ μ . R_0 = 1 + \frac{r\max_{a,b}|\widetilde{A}_{ab}|}{\mu}. R 0 = 1 + μ r max a , b ∣ A ab ∣ . Its bit length is polynomial. To bisect it at a rational z z z A ~ − z M \widetilde{A}-zM A − z M a ≠ 0 a\ne0 a = 0
( 0 a a 0 ) , \begin{pmatrix}0 & a\\a& 0\end{pmatrix}, ( 0 a a 0 ) , which contributes one positive and one negative direction. A zero remaining matrix contributes only zero directions. These operations give the full inertia. After denominators are cleared, each Schur entry is a ratio of bordered minors; determinant bounds again give polynomial bit growth. Thus this test is a polynomial-time exact rational calculation, including the case of zero eigenvalues.
Because M M M A ~ − z M \widetilde{A}-zM A − z M z ≤ λ ~ z\le\widetilde{\lambda} z ≤ λ [ ℓ , u ] [\ell,u] [ ℓ , u ] λ ~ \widetilde{\lambda} λ ε D / 8 \varepsilon_D/8 ε D /8
The fixed lower bound on D D D C e − 1.16 D ≤ ε D / 8 Ce^{-1.16D}\le\varepsilon_D/8 C e − 1.16 D ≤ ε D /8 ( 8 / 5 ) log 2 < 1.16 (8/5)\log2<1.16 ( 8/5 ) log 2 < 1.16
λ i ∈ [ ℓ − ε D 16 − ε D 8 , u + ε D 16 ] . (57) \lambda_i\in\left[\ell-\frac{\varepsilon_D}{16}-\frac{\varepsilon_D}{8},u+\frac{\varepsilon_D}{16}\right]. \tag*{(57)} λ i ∈ [ ℓ − 16 ε D − 8 ε D , u + 16 ε D ] . (57) Its width is at most 3 ε D / 8 3\varepsilon_D/8 3 ε D /8
All dimensions, operation counts, and working bit lengths are polynomial in D D D m m m B B B D D D
Simultaneous rational tuning
The energy oracle now supplies certified intervals, while Theorem 3.2 supplies the monotonicity and weak coupling of the tuning parameters. Combining these facts gives a rational algorithm: each coordinate is bracketed separately, and simultaneous updates contract. The stopping rule below uses an interval containing the target energy; it never asks for the exact sign of an unknown eigenvalue difference.
Proposition 3.7 (Tuning with fixed rounded charges). At any fixed rational centers satisfying the hypotheses of this section, and with the integer charges in (23) already fixed, one can compute rational ρ 1 , … , ρ m ∈ [ 1 , 4 ] \rho_1,\ldots,\rho_m\in[1,4] ρ 1 , … , ρ m ∈ [ 1 , 4 ]
∣ λ i + 1 2 ∣ ≤ e − 1.01 D ( 1 ≤ i ≤ m ) . (58) \left|\lambda_i+\frac{1}{2}\right|\le e^{-1.01D}\qquad(1\le i\le m). \tag*{(58)} λ i + 2 1 ≤ e − 1.01 D ( 1 ≤ i ≤ m ) . (58) The algorithm is deterministic and polynomial in D D D m m m O ( D ) O(D) O ( D ) Proof. For frozen parameters other than ρ i \rho_i ρ i λ i = − 1 / 2 \lambda_i=-1/2 λ i = − 1/2 [ 1 , 4 ] [1,4] [ 1 , 4 ] T i ( ρ ) T_i(\rho) T i ( ρ ) T = ( T 1 , … , T m ) T=(T_1,\ldots,T_m) T = ( T 1 , … , T m )
∥ T ( ρ ) − T ( ρ ~ ) ∥ ∞ ≤ θ ∥ ρ − ρ ~ ∥ ∞ , θ = C m / D 2 ≤ 1 / 2. (59) \lVert T(\rho)-T(\widetilde{\rho})\rVert_\infty\leq\theta\lVert\rho-\widetilde{\rho}\rVert_\infty,\qquad\theta=Cm/D^2\leq1/2. \tag*{(59)} ∥ T ( ρ ) − T ( ρ ) ∥ ∞ ≤ θ ∥ ρ − ρ ∥ ∞ , θ = C m / D 2 ≤ 1/2. (59) Thus T T T ρ ∗ \rho^* ρ ∗
We implement each coordinate update with a certified scalar bisection. Start with [ 1 , 4 ] [1,4] [ 1 , 4 ] − 1 / 2 -1/2 − 1/2 − 1 / 2 -1/2 − 1/2 − 1 / 2 -1/2 − 1/2 − 1 / 2 -1/2 − 1/2 ε D \varepsilon_D ε D C ε D / q ′ C\varepsilon_D/q' C ε D / q ′ J = ⌈ 4 D ⌉ J=\lceil4D\rceil J = ⌈ 4 D ⌉ 3 2 − J 3\,2^{-J} 3 2 − J q ′ ≤ 1 q'\leq1 q ′ ≤ 1 2 − 4 D ≪ ε D 2^{-4D}\ll\varepsilon_D 2 − 4 D ≪ ε D
∥ T b ( ρ ) − T ( ρ ) ∥ ∞ ≤ C ε D / q ′ . (60) \lVert T^b(\rho)-T(\rho)\rVert_\infty\leq C\varepsilon_D/q'. \tag*{(60)} ∥ T b ( ρ ) − T ( ρ ) ∥ ∞ ≤ C ε D / q ′ . (60) This comparison-or-stop rule never requests an exact sign of an unknown irrational eigenvalue at a putative zero.
Start with all coordinates equal to two and perform J J J e j e_j e j ρ ∗ \rho^* ρ ∗
e j + 1 ≤ 1 2 e j + C ε D / q ′ , e J ≤ 3 2 − J + 2 C ε D / q ′ . e_{j+1}\leq\frac{1}{2}e_j+C\varepsilon_D/q',\qquad e_J\leq3\,2^{-J}+2C\varepsilon_D/q'. e j + 1 ≤ 2 1 e j + C ε D / q ′ , e J ≤ 3 2 − J + 2 C ε D / q ′ . The own-coordinate upper slope and the other-coordinate variation bound together give
∣ λ i ( ρ ) − λ i ( ρ ∗ ) ∣ ≤ C q ′ ∥ ρ − ρ ∗ ∥ ∞ . \lvert\lambda_i(\rho)-\lambda_i(\rho^*)\rvert\leq Cq'\lVert\rho-\rho^*\rVert_\infty. ∣ λ i ( ρ ) − λ i ( ρ ∗ )∣ ≤ C q ′ ∥ ρ − ρ ∗ ∥ ∞ . Consequently the output has
∣ λ i + 1 / 2 ∣ ≤ C q ′ 2 − J + C ε D ≤ e − 1.01 D \lvert\lambda_i+1/2\rvert\leq Cq'2^{-J}+C\varepsilon_D\leq e^{-1.01D} ∣ λ i + 1/2 ∣ ≤ C q ′ 2 − J + C ε D ≤ e − 1.01 D for large enough D D D ( 8 / 5 ) log 2 > 1.1 > 1.01 (8/5)\log2>1.1>1.01 ( 8/5 ) log 2 > 1.1 > 1.01
Each scalar bisection restarts from the rational interval [ 1 , 4 ] [1,4] [ 1 , 4 ] O ( D ) O(D) O ( D ) O ( D ) O(D) O ( D ) O ( m D 2 ) O(mD^2) O ( m D 2 ) O ( D ) O(D) O ( D )
Relative tails and computable hopping calibration
The absolute orbital estimate of Proposition 3.1 does not determine an overlap of order e − D e^{-D} e − D D D D ρ i ∈ [ 1 , 4 ] \rho_i\in[1,4] ρ i ∈ [ 1 , 4 ] χ \chi χ
We use the positions and operators from Section 3. In particular,
X i = D p i + s i , max i ∣ s i ∣ ≤ S , ∣ X i − X j ∣ ≥ D − 2 S ( i ≠ j ) . X_i = Dp_i+s_i,\qquad\max_i |s_i|\le S,\qquad|X_i-X_j|\ge D-2S\quad(i\ne j). X i = D p i + s i , i max ∣ s i ∣ ≤ S , ∣ X i − X j ∣ ≥ D − 2 S ( i = j ) . We take D D D m + S ≤ D 1 / 4 m+S\le D^{1/4} m + S ≤ D 1/4
Relative comparison of single-well tails
For r i = ∣ y − X i ∣ r_i=|y-X_i| r i = ∣ y − X i ∣
F i ( y ) = r i ∫ 0 1 W i ( X i + t ( y − X i ) ) d t , F i ( X i ) = 0. (61) F_i(y)=r_i\int_0^1 W_i\bigl(X_i+t(y-X_i)\bigr)\,\mathrm{d}t,\qquad F_i(X_i)=0. \tag*{(61)} F i ( y ) = r i ∫ 0 1 W i ( X i + t ( y − X i ) ) d t , F i ( X i ) = 0. (61) The integral follows the entire segment from X i X_i X i y y y
ε r e l = m log D + m 2 D . \varepsilon_{\mathrm{rel}}=\frac{m\log D+m^2}{D}. ε rel = D m log D + m 2 . Proposition 4.1 (Uniform relative tail). Suppose the actual rounded charges and the parameters ρ i \rho_i ρ i ∣ λ i + 1 / 2 ∣ ≤ e − 1.01 D |\lambda_i+1/2|\le e^{-1.01D} ∣ λ i + 1/2∣ ≤ e − 1.01 D r i ≤ 2 D r_i\le2D r i ≤ 2 D
ψ i ( y ) = π − 1 / 2 e − r i + F i ( y ) ( 1 + O ( ε r e l ) ) . \psi_i(y)=\pi^{-1/2}e^{-r_i+F_i(y)}\bigl(1+O(\varepsilon_{\mathrm{rel}})\bigr). ψ i ( y ) = π − 1/2 e − r i + F i ( y ) ( 1 + O ( ε rel ) ) . Proof. Fix a site, suppress its subscript, and write r = ∣ y − X ∣ r=|y-X| r = ∣ y − X ∣ ω = ( y − X ) / r \omega=(y-X)/r ω = ( y − X ) / r r > 0 r>0 r > 0
∥ W ∥ ∞ ≤ C m D , ∥ ∇ W ∥ ∞ ≤ C m D 2 , ∥ D 2 W ∥ ∞ ≤ C m D 3 . \|W\|_\infty\le\frac{Cm}{D},\qquad\|\nabla W\|_\infty\le\frac{Cm}{D^2},\qquad\|D^2W\|_\infty\le\frac{Cm}{D^3}. ∥ W ∥ ∞ ≤ D C m , ∥∇ W ∥ ∞ ≤ D 2 C m , ∥ D 2 W ∥ ∞ ≤ D 3 C m . Set A ( y ) = ∫ 0 1 W ( X + t ( y − X ) ) d t A(y)=\int_0^1W\bigl(X+t(y-X)\bigr)\,\mathrm{d}t A ( y ) = ∫ 0 1 W ( X + t ( y − X ) ) d t F = r A F=rA F = r A 0 < r ≤ 3 D 0<r\le3D 0 < r ≤ 3 D
∇ F = ω A + r ∫ 0 1 t ∇ W ( X + t ( y − X ) ) d t , \nabla F=\omega A+r\int_0^1t\nabla W\bigl(X+t(y-X)\bigr)\,\mathrm{d}t, ∇ F = ω A + r ∫ 0 1 t ∇ W ( X + t ( y − X ) ) d t , Δ F = 2 A r + 2 ω ⋅ ∫ 0 1 t ∇ W ( X + t ( y − X ) ) d t + r ∫ 0 1 t 2 Δ W ( X + t ( y − X ) ) d t . \Delta F=\frac{2A}{r}+2\omega\cdot\int_0^1t\nabla W\bigl(X+t(y-X)\bigr)\,\mathrm{d}t+r\int_0^1t^2\Delta W\bigl(X+t(y-X)\bigr)\,\mathrm{d}t. Δ F = r 2 A + 2 ω ⋅ ∫ 0 1 t ∇ W ( X + t ( y − X ) ) d t + r ∫ 0 1 t 2 Δ W ( X + t ( y − X ) ) d t . Consequently,
∣ ∇ F ∣ ≤ C m D , ∣ Δ F ∣ ≤ C ( m D r + m D 2 ) . |\nabla F|\le\frac{Cm}{D},\qquad|\Delta F|\le C\left(\frac{m}{Dr}+\frac{m}{D^2}\right). ∣∇ F ∣ ≤ D C m , ∣Δ F ∣ ≤ C ( D r m + D 2 m ) . Also F ( r , ω ) = ∫ 0 r W ( X + t ω ) d t F(r,\omega)=\int_0^rW(X+t\omega)\,\mathrm{d}t F ( r , ω ) = ∫ 0 r W ( X + t ω ) d t ∂ r F = W ( y ) \partial_rF=W(y) ∂ r F = W ( y ) f = π − 1 / 2 e − r + F f=\pi^{-1/2}e^{-r+F} f = π − 1/2 e − r + F 10 ≤ r ≤ 3 D 10\le r\le3D 10 ≤ r ≤ 3 D
( h i − λ i ) f f = − ( λ i + 1 / 2 ) + 1 r − V i − 1 2 ( Δ F + ∣ ∇ F ∣ 2 ) , (62) \frac{(h_i-\lambda_i)f}{f}=-(\lambda_i+1/2)+\frac{1}{r}-V_i-\frac{1}{2}\bigl(\Delta F+|\nabla F|^2\bigr), \tag*{(62)} f ( h i − λ i ) f = − ( λ i + 1/2 ) + r 1 − V i − 2 1 ( Δ F + ∣∇ F ∣ 2 ) , (62) ∣ ( h i − λ i ) f f ∣ ≤ C ( m D r + m 2 D 2 ) . (63) \left|\frac{(h_i-\lambda_i)f}{f}\right|\le C\left(\frac{m}{Dr}+\frac{m^2}{D^2}\right). \tag*{(63)} f ( h i − λ i ) f ≤ C ( D r m + D 2 m 2 ) . (63) In the first equality the term ∂ r F \partial_r F ∂ r F W i W_i W i ρ i ≤ 4 \rho_i \le4 ρ i ≤ 4 r ≥ 10 r \ge10 r ≥ 10
∣ 1 r − V i ( y ) ∣ ≤ ∣ 1 − q ∣ r + q ′ r − 4 ≤ C m D r . \left|\frac{1}{r}-V_i(y)\right| \le\frac{|1-q|}{r}+\frac{q'}{r-4}\le\frac{Cm}{Dr}. r 1 − V i ( y ) ≤ r ∣1 − q ∣ + r − 4 q ′ ≤ D r C m . The charge-rounding errors and e − 1.01 D e^{-1.01D} e − 1.01 D (63) . Choose a sufficiently large fixed constant A 0 A_0 A 0
g ( r ) = A 0 [ m D ( 1 + log ( r / 10 ) ) + m 2 r D 2 ] . (64) g(r)=A_0\left[\frac{m}{D}(1+\log(r/10))+\frac{m^2r}{D^2}\right]. \tag*{(64)} g ( r ) = A 0 [ D m ( 1 + log ( r /10 )) + D 2 m 2 r ] . (64) Then
g ′ = A 0 ( m D r + m 2 D 2 ) , Δ g = A 0 ( m D r 2 + 2 m 2 D 2 r ) , 0 ≤ Δ g ≤ 2 g ′ r . g'=A_0\left(\frac{m}{Dr}+\frac{m^2}{D^2}\right),\qquad\Delta g=A_0\left(\frac{m}{Dr^2}+\frac{2m^2}{D^2r}\right),\qquad0\le\Delta g\le\frac{2g'}{r}. g ′ = A 0 ( D r m + D 2 m 2 ) , Δ g = A 0 ( D r 2 m + D 2 r 2 m 2 ) , 0 ≤ Δ g ≤ r 2 g ′ . Replacing f f f f e ± g fe^{\pm g} f e ± g
± ( 1 − ∂ r F ) g ′ + 1 2 Δ g − 1 2 ( g ′ ) 2 . (65) \pm(1-\partial_r F)g'+\frac{1}{2}\Delta g-\frac{1}{2}(g')^2. \tag*{(65)} ± ( 1 − ∂ r F ) g ′ + 2 1 Δ g − 2 1 ( g ′ ) 2 . (65) Since r ≥ 10 r\ge10 r ≥ 10 ∂ r F = O ( m / D ) \partial_r F=O(m/D) ∂ r F = O ( m / D ) g ′ g' g ′ D D D c g ′ cg' c g ′ − c g ′ -cg' − c g ′ A 0 A_0 A 0 (63) . Thus f e g fe^g f e g f e − g fe^{-g} f e − g
At r = 10 r=10 r = 10 ψ i = π − 1 / 2 e − 10 + O ( m / D ) \psi_i=\pi^{-1/2}e^{-10}+O(m/D) ψ i = π − 1/2 e − 10 + O ( m / D ) F = O ( m / D ) F=O(m/D) F = O ( m / D ) e − 10 e^{-10} e − 10 A 0 A_0 A 0 f e − g ≤ ψ i ≤ f e g fe^{-g}\le\psi_i\le fe^g f e − g ≤ ψ i ≤ f e g r = 3 D r=3D r = 3 D
∣ ψ i ∣ ≤ C e − 0.99 ( 3 − 0.01 ) D = C e − 2.9601 D . |\psi_i|\le Ce^{-0.99(3-0.01)D}=Ce^{-2.9601D}. ∣ ψ i ∣ ≤ C e − 0.99 ( 3 − 0.01 ) D = C e − 2.9601 D . Moreover 0 ≤ F ≤ C m 0\le F\le Cm 0 ≤ F ≤ C m κ = e − 2.5 D \kappa=e^{-2.5D} κ = e − 2.5 D
f + = f e g + κ , f − = f e − g − κ f_+=fe^g+\kappa,\qquad f_-=fe^{-g}-\kappa f + = f e g + κ , f − = f e − g − κ satisfy f − ≤ ψ i ≤ f + f_-\le\psi_i\le f_+ f − ≤ ψ i ≤ f + D D D
− V i − W i − λ i ≥ 1 2 − 1 10 − O ( m / D ) ≥ 1 4 . -V_i-W_i-\lambda_i\ge\frac{1}{2}-\frac{1}{10}-O(m/D)\ge\frac{1}{4}. − V i − W i − λ i ≥ 2 1 − 10 1 − O ( m / D ) ≥ 4 1 . Adding the positive constant κ \kappa κ L 2 L^2 L 2
On 10 ≤ r ≤ 2 D 10\le r\le2D 10 ≤ r ≤ 2 D g ( r ) = O ( ε r e l ) g(r)=O(\varepsilon_{\mathrm{rel}}) g ( r ) = O ( ε rel ) F ≥ 0 F\ge0 F ≥ 0
κ f ≤ π e − D / 2 . \frac{\kappa}{f}\le\sqrt{\pi}\,e^{-D/2}. f κ ≤ π e − D /2 . The comparison inequalities imply (4.3) on this region. On r ≤ 10 r\le10 r ≤ 10 F = O ( m / D ) F=O(m/D) F = O ( m / D )
Surrogate fields and positive hopping integrals
We first identify the attraction integral to be calibrated. Write V k h o l e = ( 1 − χ k ) V k V_k^{\mathrm{hole}}=(1-\chi_k)V_k V k hole = ( 1 − χ k ) V k k k k ( i , j ) (i,j) ( i , j )
B i j = ⟨ ψ j , V j h o l e ψ i ⟩ . (66) B_{ij}=\langle\psi_j,V_j^{\mathrm{hole}}\psi_i\rangle. \tag*{(66)} B ij = ⟨ ψ j , V j hole ψ i ⟩ . (66) The identity
h = h i − ∑ k ≠ i V k h o l e h=h_i-\sum_{k\ne i}V_k^{\mathrm{hole}} h = h i − k = i ∑ V k hole shows that the hole at j j j − B i j -B_{ij} − B ij ⟨ ψ j , ( h + 1 / 2 ) ψ i ⟩ \langle\psi_j,(h+1/2)\psi_i\rangle ⟨ ψ j , ( h + 1/2 ) ψ i ⟩
The relative tail formula still contains the secondary parameters, which have not yet been computed when the primary centers must be placed. We replace its capped field by an explicit one depending only on the primary centers. The replacement error is relative and uniform in the entire tuning cube; this is what makes the construction order possible.
Define
W i ∗ ( y ) = ∑ k ≠ i χ ( ∣ y − X k ∣ / D ) ∣ y − X k ∣ , F i ∗ ( y ) = r i ∫ 0 1 W i ∗ ( X i + t ( y − X i ) ) d t . (67) W_i^*(y)=\sum_{k\ne i}\frac{\chi(|y-X_k|/D)}{|y-X_k|},\qquad F_i^*(y)=r_i\int_0^1W_i^*(X_i+t(y-X_i))\,\mathrm{d}t. \tag*{(67)} W i ∗ ( y ) = k = i ∑ ∣ y − X k ∣ χ ( ∣ y − X k ∣/ D ) , F i ∗ ( y ) = r i ∫ 0 1 W i ∗ ( X i + t ( y − X i )) d t . (67) Each summand is extended by zero through its cutoff hole. In particular the apparent quotient at y = X k y=X_k y = X k
Lemma 4.2 (Uniform capped kernels and surrogate replacement). The zero-extended kernels
G D ( z ) = χ ( ∣ z ∣ / D ) ∣ z ∣ , J D , ρ ( z ) = χ ( ∣ z ∣ / D ) ∣ z − ρ e z ∣ G_D(z)=\frac{\chi(|z|/D)}{|z|},\qquad J_{D,\rho}(z)=\frac{\chi(|z|/D)}{|z-\rho e_z|} G D ( z ) = ∣ z ∣ χ ( ∣ z ∣/ D ) , J D , ρ ( z ) = ∣ z − ρ e z ∣ χ ( ∣ z ∣/ D ) are globally C 2 C^2 C 2 1 ≤ ρ ≤ 4 1\le\rho\le4 1 ≤ ρ ≤ 4 α \alpha α ∣ α ∣ ≤ 2 |\alpha|\le2 ∣ α ∣ ≤ 2
∥ ∂ α G D ∥ ∞ + ∥ ∂ α J D , ρ ∥ ∞ ≤ C D − 1 − ∣ α ∣ . (68) \|\partial^\alpha G_D\|_\infty+\|\partial^\alpha J_{D,\rho}\|_\infty\le CD^{-1-|\alpha|}. \tag*{(68)} ∥ ∂ α G D ∥ ∞ + ∥ ∂ α J D , ρ ∥ ∞ ≤ C D − 1 − ∣ α ∣ . (68) Uniformly over all allowed secondary positions and rounded charges,
∥ W i − W i ∗ ∥ ∞ ≤ C m 2 D 2 , ∣ F i ( y ) − F i ∗ ( y ) ∣ ≤ C m 2 D ( r i ≤ 3 D ) . (69) \|W_i-W_i^*\|_\infty\le\frac{Cm^2}{D^2},\qquad|F_i(y)-F_i^*(y)|\le\frac{Cm^2}{D}\quad(r_i\le3D). \tag*{(69)} ∥ W i − W i ∗ ∥ ∞ ≤ D 2 C m 2 , ∣ F i ( y ) − F i ∗ ( y ) ∣ ≤ D C m 2 ( r i ≤ 3 D ) . (69) Consequently, for tuned wells the relative formula (4.3) holds with F i ∗ F_i^* F i ∗ in place of F i F_i F i and the same error bound O ( ε r e l ) O(\varepsilon_{\mathrm{rel}}) O ( ε rel )
Proof. For large D D D ∣ z ∣ < D / 16 |z|<D/16 ∣ z ∣ < D /16 ∣ z ∣ ≥ D / 16 |z|\ge D/16 ∣ z ∣ ≥ D /16 ∣ z − ρ e z ∣ ≥ ∣ z ∣ / 2 |z-\rho e_z|\ge|z|/2 ∣ z − ρ e z ∣ ≥ ∣ z ∣/2 z = D ζ z=D\zeta z = D ζ D − 1 D^{-1} D − 1 ( ρ / D ) e z (\rho/D)e_z ( ρ / D ) e z χ \chi χ C 2 C^2 C 2 χ \chi χ (68) .
Since ∣ q − 1 ∣ + q ′ ≤ C m / D |q-1|+q'\le Cm/D ∣ q − 1∣ + q ′ ≤ C m / D
W i − W i ∗ = ∑ k ≠ i [ ( q − 1 ) G D ( y − X k ) + q ′ J D , ρ k ( y − X k ) ] W_i-W_i^*=\sum_{k\ne i}\left[(q-1)G_D(y-X_k)+q'J_{D,\rho_k}(y-X_k)\right] W i − W i ∗ = k = i ∑ [ ( q − 1 ) G D ( y − X k ) + q ′ J D , ρ k ( y − X k ) ] has sup norm O ( m 2 / D 2 ) O(m^2/D^2) O ( m 2 / D 2 ) 3 D 3D 3 D (69) . Exponentiating this bound in Proposition 4.1 proves the final assertion. None of the kernel estimates require tuning; they hold uniformly on the whole parameter cube. ∎
For an oriented contact ( i , j ) (i,j) ( i , j ) B i j B_{ij} B ij
T i j ( X ) = 1 π ∫ ∣ v ∣ ≤ D / 8 1 − χ ( ∣ v ∣ / D ) ∣ v ∣ e L X ( v ) d 3 v , (70) T_{ij}(X)=\frac{1}{\pi}\int_{\lvert v\rvert\le D/8}\frac{1-\chi(\lvert v\rvert/D)}{\lvert v\rvert}e^{L_X(v)}\,\mathrm{d}^3v, \tag*{(70)} T ij ( X ) = π 1 ∫ ∣ v ∣ ≤ D /8 ∣ v ∣ 1 − χ (∣ v ∣ / D ) e L X ( v ) d 3 v , (70) L X ( v ) = − ∣ X j + v − X i ∣ − ∣ v ∣ + F i ∗ ( X j + v ) + F j ∗ ( X j + v ) . (71) L_X(v)=-\lvert X_j+v-X_i\rvert-\lvert v\rvert+F_i^*(X_j+v)+F_j^*(X_j+v). \tag*{(71)} L X ( v ) = − ∣ X j + v − X i ∣ − ∣ v ∣ + F i ∗ ( X j + v ) + F j ∗ ( X j + v ) . (71) The fields in this definition use all centers in X X X ∣ v ∣ − 1 \lvert v\rvert^{-1} ∣ v ∣ − 1
Lemma 4.3 (Comparison with the uncapped attraction). For tuned wells and every contact,
B i j = T i j ( X ) ( 1 + O ( ε r e l ) ) . (72) B_{ij}=T_{ij}(X)(1+O(\varepsilon_{\mathrm{rel}})). \tag*{(72)} B ij = T ij ( X ) ( 1 + O ( ε rel )) . (72) The constants are uniform in ρ j ∈ [ 1 , 4 ] \rho_j\in[1,4] ρ j ∈ [ 1 , 4 ]
Proof. On the support of V j h o l e V_j^{\mathrm{hole}} V j hole y = X j + v y=X_j+v y = X j + v ∣ v ∣ ≤ D / 8 \lvert v\rvert\le D/8 ∣ v ∣ ≤ D /8 ∣ y − X i ∣ ≤ D + 2 S + D / 8 < 2 D \lvert y-X_i\rvert\le D+2S+D/8<2D ∣ y − X i ∣ ≤ D + 2 S + D /8 < 2 D π − 1 e L X ( v ) \pi^{-1}e^{L_X(v)} π − 1 e L X ( v ) 1 + O ( ε r e l ) 1+O(\varepsilon_{\mathrm{rel}}) 1 + O ( ε rel ) q = 1 + O ( m / D ) q=1+O(m/D) q = 1 + O ( m / D )
For ∣ v ∣ > 10 \lvert v\rvert>10 ∣ v ∣ > 10
∣ v ∣ ∣ v − ρ j e z ∣ ≤ ∣ v ∣ ∣ v ∣ − 4 ≤ 5 3 . \frac{\lvert v\rvert}{\lvert v-\rho_j e_z\rvert}\le\frac{\lvert v\rvert}{\lvert v\rvert-4}\le\frac{5}{3}. ∣ v − ρ j e z ∣ ∣ v ∣ ≤ ∣ v ∣ − 4 ∣ v ∣ ≤ 3 5 . For ∣ v ∣ ≤ 10 \lvert v\rvert\le10 ∣ v ∣ ≤ 10 D D D F ∗ F^* F ∗ L X L_X L X ∣ v ∣ \lvert v\rvert ∣ v ∣ e L X ( v ) e^{L_X(v)} e L X ( v ) e L X ( 0 ) e^{L_X(0)} e L X ( 0 )
∫ ∣ v ∣ ≤ 10 d 3 v ∣ v − ρ j e z ∣ ≤ ∫ ∣ w ∣ ≤ 14 d 3 w ∣ w ∣ < ∞ , ∫ ∣ v ∣ ≤ 1 d 3 v ∣ v ∣ > 0. \int_{\lvert v\rvert\le10}\frac{\mathrm{d}^3v}{\lvert v-\rho_j e_z\rvert}\le\int_{\lvert w\rvert\le14}\frac{\mathrm{d}^3w}{\lvert w\rvert}<\infty,\qquad
\int_{\lvert v\rvert\le1}\frac{\mathrm{d}^3v}{\lvert v\rvert}>0. ∫ ∣ v ∣ ≤ 10 ∣ v − ρ j e z ∣ d 3 v ≤ ∫ ∣ w ∣ ≤ 14 ∣ w ∣ d 3 w < ∞ , ∫ ∣ v ∣ ≤ 1 ∣ v ∣ d 3 v > 0. The secondary-kernel integral on the inner ball is therefore at most a fixed multiple of the primary-kernel integral. Combining the two regions and multiplying by q ′ = O ( m / D ) q'=O(m/D) q ′ = O ( m / D ) O ( m / D ) O(m/D) O ( m / D ) (72) .
Lemma 4.4 (Size of the reference integrals). Let X 0 = ( D p i ) i X^0=(Dp_i)_i X 0 = ( D p i ) i
c e − D ≤ T i j ( X 0 ) ≤ C D 2 e − D + C m . (73) ce^{-D}\le T_{ij}(X^0)\le CD^2e^{-D+Cm}. \tag*{(73)} c e − D ≤ T ij ( X 0 ) ≤ C D 2 e − D + C m . (73) Proof. Put u = u i j = p j − p i u=u_{ij}=p_j-p_i u = u ij = p j − p i ∣ u ∣ = 1 \lvert u\rvert=1 ∣ u ∣ = 1 ∣ D u + v ∣ + ∣ v ∣ ≥ D \lvert Du+v\rvert+\lvert v\rvert\ge D ∣ D u + v ∣ + ∣ v ∣ ≥ D C m Cm C m (68) and its ray length. The integral of ∣ v ∣ − 1 \lvert v\rvert^{-1} ∣ v ∣ − 1 D / 8 D/8 D /8 O ( D 2 ) O(D^2) O ( D 2 ) ∣ v ∣ ≤ 1 \lvert v\rvert\le1 ∣ v ∣ ≤ 1 ∣ D u + v ∣ + ∣ v ∣ ≤ D + 2 \lvert Du+v\rvert+\lvert v\rvert\le D+2 ∣ D u + v ∣ + ∣ v ∣ ≤ D + 2 2 e − D − 2 2e^{-D-2} 2 e − D − 2 D D D
Sensitivity to displacement
Lemma 4.5 (Uniform displacement law). Let X i = D p i + s i X_i = Dp_i + s_i X i = D p i + s i ∣ s i ∣ ≤ S |s_i| \le S ∣ s i ∣ ≤ S c 1 > 0 c_1 > 0 c 1 > 0
T i j ( X ) T i j ( X 0 ) = e − u i j ⋅ ( s j − s i ) ( 1 + O ( ε d i s p ) ) , (74) \frac{T_{ij}(X)}{T_{ij}(X^0)} = e^{-u_{ij}\cdot(s_j-s_i)}\left(1+O(\varepsilon_{\mathrm{disp}})\right), \tag*{(74)} T ij ( X 0 ) T ij ( X ) = e − u ij ⋅ ( s j − s i ) ( 1 + O ( ε disp ) ) , (74) ε d i s p = m S D + S 2 D + S D − 1 / 4 + e − c 1 D . (75) \varepsilon_{\mathrm{disp}} = \frac{mS}{D}+\frac{S^2}{D}+SD^{-1/4}+e^{-c_1\sqrt{D}}. \tag*{(75)} ε disp = D m S + D S 2 + S D − 1/4 + e − c 1 D . (75) In particular the coefficient of the projected displacement in the leading exponent is one.
Proof. Compare the two integrals at the same v v v F i ∗ ( X j + v ) F_i^*(X_j+v) F i ∗ ( X j + v ) D p i + t ( D p j + v − D p i ) Dp_i+t(Dp_j+v-Dp_i) D p i + t ( D p j + v − D p i ) ( 1 − t ) s i + t s j (1-t)s_i+ts_j ( 1 − t ) s i + t s j S S S S S S (68) therefore changes the field along the path by at most C m S / D 2 C mS/D^2 C m S / D 2 O ( D ) O(D) O ( D ) 2 S 2S 2 S O ( m / D ) O(m/D) O ( m / D )
∣ F i ∗ ( X j + v ; X ) − F i ∗ ( D p j + v ; X 0 ) ∣ ≤ C m S D . (76) \left|F_i^*(X_j+v;X)-F_i^*(Dp_j+v;X^0)\right|\le\frac{C mS}{D}. \tag*{(76)} F i ∗ ( X j + v ; X ) − F i ∗ ( D p j + v ; X 0 ) ≤ D C m S . (76) The same bound holds for the F j ∗ F_j^* F j ∗ ∣ v ∣ |v| ∣ v ∣
Write u = u i j u=u_{ij} u = u ij δ = s j − s i \delta=s_j-s_i δ = s j − s i v = z u + w v=zu+w v = z u + w w ⊥ u w\perp u w ⊥ u ∣ v ∣ ≤ D / 8 |v|\le D/8 ∣ v ∣ ≤ D /8 ∣ D u + v ∣ ≥ 7 D / 8 |Du+v|\ge7D/8 ∣ D u + v ∣ ≥ 7 D /8 D u + v Du+v D u + v D u + v + δ Du+v+\delta D u + v + δ O ( 1 / D ) O(1/D) O ( 1/ D ) D u + v Du+v D u + v u u u O ( ∣ w ∣ / D ) O(|w|/D) O ( ∣ w ∣/ D )
∣ D u + v + δ ∣ − ∣ D u + v ∣ = u ⋅ δ + O ( S ∣ w ∣ D + S 2 D ) . (77) |Du+v+\delta|-|Du+v|=u\cdot\delta+O\left(\frac{S|w|}{D}+\frac{S^2}{D}\right). \tag*{(77)} ∣ D u + v + δ ∣ − ∣ D u + v ∣ = u ⋅ δ + O ( D S ∣ w ∣ + D S 2 ) . (77) This estimate includes the whole backward segment v = − t u v=-tu v = − t u 0 ≤ t ≤ D / 8 0\le t\le D/8 0 ≤ t ≤ D /8 D u + v = ( D − t ) u Du+v=(D-t)u D u + v = ( D − t ) u u u u O ( S ) O(S) O ( S )
On the region ∣ w ∣ ≤ D 3 / 4 |w|\le D^{3/4} ∣ w ∣ ≤ D 3/4 (76) and (77) show that
L X ( v ) − L X 0 ( v ) + u ⋅ δ = O ( m S D + S 2 D + S D − 1 / 4 ) . (78) L_X(v)-L_{X^0}(v)+u\cdot\delta=O\left(\frac{mS}{D}+\frac{S^2}{D}+SD^{-1/4}\right). \tag*{(78)} L X ( v ) − L X 0 ( v ) + u ⋅ δ = O ( D m S + D S 2 + S D − 1/4 ) . (78) The nonnegative integration weight is identical in the two integrals, so exponentiation controls their contributions on this region by the same relative factor.
On the remaining region ∣ w ∣ > D 3 / 4 |w|>D^{3/4} ∣ w ∣ > D 3/4
∣ D u + v ∣ + ∣ v ∣ − D ≥ z + ∣ v ∣ = ∣ w ∣ 2 ∣ v ∣ − z ≥ 4 ∣ w ∣ 2 D ≥ 4 D . (79) |Du+v|+|v|-D\ge z+|v|=\frac{|w|^2}{|v|-z}\ge\frac{4|w|^2}{D}\ge4\sqrt{D}. \tag*{(79)} ∣ D u + v ∣ + ∣ v ∣ − D ≥ z + ∣ v ∣ = ∣ v ∣ − z ∣ w ∣ 2 ≥ D 4∣ w ∣ 2 ≥ 4 D . (79) The identity is used only for w ≠ 0 w\ne0 w = 0 2 ∣ v ∣ ≤ D / 4 2|v|\le D/4 2∣ v ∣ ≤ D /4 C m Cm C m 2 S 2S 2 S ∣ u ⋅ δ ∣ ≤ 2 S |u \cdot\delta| \le2S ∣ u ⋅ δ ∣ ≤ 2 S e − D e^{-D} e − D e − D − u ⋅ δ e^{-D-u\cdot\delta} e − D − u ⋅ δ
C D 2 exp [ − c D + C ( m + S ) ] . (80) CD^2 \exp\left[-c\sqrt{D}+C(m+S)\right]. \tag*{(80)} C D 2 exp [ − c D + C ( m + S ) ] . (80) By (73) , these are also bounds on their contributions relative to the reference integral, with the appropriate displacement factor. The assumption m + S ≤ D 1 / 4 m+S \le D^{1/4} m + S ≤ D 1/4 D D D e C m e^{Cm} e C m e − c 1 D e^{-c_1\sqrt{D}} e − c 1 D (78) , and using smallness of its right-hand side, proves the lemma.
Deterministic relative quadrature
We have reduced hopping calibration to positive integrals whose size is exponentially small. Direct absolute quadrature at that scale would obscure the complexity bound. Instead we control the oscillation of the logarithm of the exponential factor, while integrating its nonnegative radial weight exactly. Positivity then converts local errors into a relative error for the whole integral.
Lemma 4.6 (Computing the positive reference integrals). For 0 < ϵ < 1 / 10 0 < \epsilon< 1/10 0 < ϵ < 1/10 I ^ i j \widehat{I}_{ij} I ij
∣ log I ^ i j e D T i j ( X 0 ) ∣ ≤ ϵ (81) \left|\log\frac{\widehat{I}_{ij}}{e^{D T_{ij}(X^0)}}\right| \le\epsilon \tag*{(81)} log e D T ij ( X 0 ) I ij ≤ ϵ (81) by a deterministic algorithm polynomial in D D D m m m ϵ − 1 \epsilon^{-1} ϵ − 1 D D D ϵ \epsilon ϵ
Proof. Let L = L X 0 L=L_{X^0} L = L X 0 (71) . On the integration ball, the two distance terms have Lipschitz constants one, and the ray formulas and (68) give a Lipschitz constant C m / D Cm/D C m / D
∣ L ( v ) − L ( v ′ ) ∣ ≤ C ∣ v − v ′ ∣ , ∣ L ( v ) ∣ ≤ C ( D + m ) . (82) |L(v)-L(v')| \le C|v-v'|,\qquad|L(v)| \le C(D+m). \tag*{(82)} ∣ L ( v ) − L ( v ′ ) ∣ ≤ C ∣ v − v ′ ∣ , ∣ L ( v ) ∣ ≤ C ( D + m ) . (82) Only the exponential factor has been included in this logarithm. No lower bound on 1 − χ 1-\chi 1 − χ log ( 1 − χ ) \log(1-\chi) log ( 1 − χ )
We describe a mesh with exact positive weights. Parameterize the unit sphere, up to a set of area zero, by
ω ( μ , φ ) = ( 1 − μ 2 cos φ , 1 − μ 2 sin φ , μ ) . \omega(\mu,\varphi)=\left(\sqrt{1-\mu^2}\cos\varphi,\sqrt{1-\mu^2}\sin\varphi,\mu\right). ω ( μ , φ ) = ( 1 − μ 2 cos φ , 1 − μ 2 sin φ , μ ) . Here − 1 ≤ μ ≤ 1 -1 \le\mu\le1 − 1 ≤ μ ≤ 1 0 ≤ φ ≤ 2 π 0 \le\varphi\le2\pi 0 ≤ φ ≤ 2 π d ω = d μ d φ d\omega=d\mu\,d\varphi d ω = d μ d φ M M M μ \mu μ N N N φ \varphi φ h h h χ ( r / D ) \chi(r/D) χ ( r / D ) I a I_a I a
w a = ∫ I a ( 1 − χ ( r / D ) ) r d r w_a=\int_{I_a}(1-\chi(r/D))r\,dr w a = ∫ I a ( 1 − χ ( r / D )) r d r is a nonnegative exact rational, computable by polynomial integration. Discard intervals with zero weight. The normalized angular weight of every angular rectangle is exactly
1 π 1 M 2 π N = 4 M N . (83) \frac{1}{\pi}\frac{1}{M}\frac{2\pi}{N}=\frac{4}{MN}. \tag*{(83)} π 1 M 1 N 2 π = M N 4 . (83) Choose one sample v a b c v_{abc} v ab c
The square-root parameterization satisfies
∣ ω ( μ , ϕ ) − ω ( μ ′ , ϕ ′ ) ∣ ≤ C ∣ μ − μ ′ ∣ 1 / 2 + C ∣ ϕ − ϕ ′ ∣ . |\omega(\mu,\phi)-\omega(\mu',\phi')| \le C|\mu-\mu'|^{1/2}+C|\phi-\phi'|. ∣ ω ( μ , ϕ ) − ω ( μ ′ , ϕ ′ ) ∣ ≤ C ∣ μ − μ ′ ∣ 1/2 + C ∣ ϕ − ϕ ′ ∣. Therefore M = O ( ( D / h ) 2 ) M=O((D/h)^2) M = O (( D / h ) 2 ) N = O ( D / h ) N=O(D/h) N = O ( D / h ) O ( h ) O(h) O ( h ) O ( ( D / h ) 4 ) O((D/h)^4) O (( D / h ) 4 )
Q = ∑ a , b , c 4 w a M N e L ( v a b c ) . Q=\sum_{a,b,c}\frac{4w_a}{MN}e^{L(v_{abc})}. Q = a , b , c ∑ M N 4 w a e L ( v ab c ) . Positivity and (82) imply
e − C h Q ≤ T i j ( X 0 ) ≤ e C h Q . (84) e^{-Ch}Q\le T_{ij}(X^0)\le e^{Ch}Q. \tag*{(84)} e − C h Q ≤ T ij ( X 0 ) ≤ e C h Q . (84) Thus a fixed multiple of ϵ \epsilon ϵ h h h e − D e^{-D} e − D
It remains to specify evaluation and bit accuracy. For a ray of length r ≤ 2 D r\le2D r ≤ 2 D t ↦ W i ∗ ( X i + t ( y − X i ) ) t\mapsto W_i^*(X_i+t(y-X_i)) t ↦ W i ∗ ( X i + t ( y − X i )) C m r / D 2 Cmr/D^2 C m r / D 2 K K K
C m r 2 D 2 K ≤ C m K . (85) \frac{Cmr^2}{D^2K}\le\frac{Cm}{K}. \tag*{(85)} D 2 K C m r 2 ≤ K C m . (85) Taking K = O ( m / ϵ ) K=O(m/\epsilon) K = O ( m / ϵ ) O ( ϵ / D ) O(\epsilon/D) O ( ϵ / D ) O ( ϵ ) O(\epsilon) O ( ϵ ) m − 1 m-1 m − 1 D D D χ ( a ) / a \chi(a)/a χ ( a ) / a [ 0 , 1 / 16 ] [0,1/16] [ 0 , 1/16 ] a = ∣ z ∣ / D a=|z|/D a = ∣ z ∣/ D
The coordinates D p i Dp_i D p i O ( ϵ ) O(\epsilon) O ( ϵ ) e O ( ϵ ) e^{O(\epsilon)} e O ( ϵ )
For the remaining arithmetic, compute the scaled sum e D Q e^D Q e D Q e D + L ( v a b c ) e^{D+L(v_{abc})} e D + L ( v ab c ) (84) , because (73) gives e D T i j ( X 0 ) ≥ 2 e − 2 e^D T_{ij}(X^0)\ge2e^{-2} e D T ij ( X 0 ) ≥ 2 e − 2 O ( D 2 ) O(D^2) O ( D 2 ) O ( D + m ) O(D+m) O ( D + m ) O ( ϵ / D 2 ) O(\epsilon/D^2) O ( ϵ / D 2 ) O ( ϵ ) O(\epsilon) O ( ϵ ) O ( D + m + log ( D / ϵ ) ) O(D+m+\log(D/\epsilon)) O ( D + m + log ( D / ϵ ))
Allocate a sufficiently small fixed fraction of ϵ \epsilon ϵ I ^ i j \widehat{I}_{ij} I ij (81) . ▫
Calibration, rounding, and orientation
The oriented attraction integrals are close to a symmetric matrix for an exact reason that is useful independently of their asymptotic formula.
Lemma 4.7 (Orientation identity). For distinct sites,
B i j − B j i = ( λ i − λ j ) ⟨ ψ j , ψ i ⟩ . (86) B_{ij}-B_{ji}=(\lambda_i-\lambda_j)\langle\psi_j,\psi_i\rangle. \tag*{(86)} B ij − B j i = ( λ i − λ j ) ⟨ ψ j , ψ i ⟩ . (86) For tuned wells and large D D D
∣ ⟨ ψ j , ψ i ⟩ ∣ ≤ e − 0.85 D , ∣ B i j − B j i ∣ ≤ 2 e − 1.86 D . (87) \lvert\langle\psi_j,\psi_i\rangle\rvert\le e^{-0.85D},\qquad\lvert B_{ij}-B_{ji}\rvert\le2e^{-1.86D}. \tag*{(87)} ∣⟨ ψ j , ψ i ⟩∣ ≤ e − 0.85 D , ∣ B ij − B j i ∣ ≤ 2 e − 1.86 D . (87) Proof. The single-well operators have common domain H 2 H^2 H 2
h i − h j = V j h o l e − V i h o l e . h_i-h_j=V_j^{\mathrm{hole}}-V_i^{\mathrm{hole}}. h i − h j = V j hole − V i hole . Pair this identity between the real eigenfunctions ψ j \psi_j ψ j ψ i \psi_i ψ i h j h_j h j (86) . All the pairings are well-defined by the Coulomb domain bounds of Proposition 3.1. Equivalently, in the two expansions of the full matrix element ⟨ ψ j , ( h + 1 / 2 ) ψ i ⟩ \langle\psi_j,(h+1/2)\psi_i\rangle ⟨ ψ j , ( h + 1/2 ) ψ i ⟩ i i i j j j
To record an explicit overlap bound, Lemma 3.3 gives
∣ ψ i ( y ) ψ j ( y ) ∣ ≤ C e 0.0198 D e − 0.99 ( r i + r j ) . \lvert\psi_i(y)\psi_j(y)\rvert\le Ce^{0.0198D}e^{-0.99(r_i+r_j)}. ∣ ψ i ( y ) ψ j ( y )∣ ≤ C e 0.0198 D e − 0.99 ( r i + r j ) . Use r i + r j ≥ D − 2 S r_i+r_j\ge D-2S r i + r j ≥ D − 2 S 0.90 0.90 0.90 e − 0.09 r i e^{-0.09r_i} e − 0.09 r i
∣ ⟨ ψ j , ψ i ⟩ ∣ ≤ C e − 0.8802 D + 1.8 S ∫ R 3 e − 0.09 ∣ y − X i ∣ d 3 y ≤ e − 0.85 D \lvert\langle\psi_j,\psi_i\rangle\rvert\le Ce^{-0.8802D+1.8S}\int_{\mathbb{R}^3}e^{-0.09\lvert y-X_i\rvert}\,d^3y\le e^{-0.85D} ∣⟨ ψ j , ψ i ⟩∣ ≤ C e − 0.8802 D + 1.8 S ∫ R 3 e − 0.09 ∣ y − X i ∣ d 3 y ≤ e − 0.85 D for sufficiently large D D D ∣ λ i − λ j ∣ ≤ 2 e − 1.01 D \lvert\lambda_i-\lambda_j\rvert\le2e^{-1.01D} ∣ λ i − λ j ∣ ≤ 2 e − 1.01 D
For a tolerance τ > 0 \tau>0 τ > 0
E c a l = m log D + m 2 D + m S D + S 2 D + S D − 1 / 4 + τ + e − c 1 D . (88) \mathcal{E}_{\mathrm{cal}}=\frac{m\log D+m^2}{D}+\frac{mS}{D}+\frac{S^2}{D}+SD^{-1/4}+\tau+e^{-c_1\sqrt{D}}. \tag*{(88)} E cal = D m log D + m 2 + D m S + D S 2 + S D − 1/4 + τ + e − c 1 D . (88) Proposition 4.8 (Deterministic hopping calibration). Let the contact layout be as in Lemma 2.4, choose one orientation per contact, and let the prescribed rational hoppings satisfy 0 < t min ≤ t i j ≤ 1 0<t_{\min}\le t_{ij}\le1 0 < t m i n ≤ t ij ≤ 1 t min t_{\min} t m i n Z ∗ Z_* Z ∗ e D / 2 ≤ Z ∗ ≤ 2 e D e^D/2\le Z_*\le2e^D e D /2 ≤ Z ∗ ≤ 2 e D τ \tau τ X i = D p i + s i X_i=Dp_i+s_i X i = D p i + s i
S : = max i ∣ s i ∣ ≤ C m ( m + log D + log ( 1 / t min ) + 1 ) + τ . (89) S:=\max_i\lvert s_i\rvert\le Cm\bigl(m+\log D+\log(1/t_{\min})+1\bigr)+\tau. \tag*{(89)} S := i max ∣ s i ∣ ≤ C m ( m + log D + log ( 1/ t m i n ) + 1 ) + τ . (89) After applying Proposition 3.7 at these final centers, the resulting orbitals satisfy, on every chosen oriented contact,
∣ Z ∗ B i j − t i j ∣ ≤ C t i j E c a l . (90) \lvert Z_*B_{ij}-t_{ij}\rvert\le Ct_{ij}\mathcal{E}_{\mathrm{cal}}. \tag*{(90)} ∣ Z ∗ B ij − t ij ∣ ≤ C t ij E cal . (90) Here D D D E c a l \mathcal{E}_{\mathrm{cal}} E cal 4 e − 0.86 D 4e^{-0.86D} 4 e − 0.86 D
Proof. Compute, to additive error at most τ / 4 \tau/4 τ /4
d i j = log ( Z ∗ T i j ( X 0 ) t i j ) , X 0 = ( D p i ) i . (91) d_{ij}=\log\left(\frac{Z_*T_{ij}(X^0)}{t_{ij}}\right), \qquad X^0=(Dp_i)_i. \tag*{(91)} d ij = log ( t ij Z ∗ T ij ( X 0 ) ) , X 0 = ( D p i ) i . (91) Lemma 4.6 supplies the requisite relative integral accuracy. One can evaluate the logarithm as
log Z ∗ − D + log ( e D T i j ( X 0 ) ) − log t i j , \log Z_* - D + \log\left(e^D T_{ij}(X^0)\right)-\log t_{ij}, log Z ∗ − D + log ( e D T ij ( X 0 ) ) − log t ij , so all elementary-function calculations have polynomial bit cost. The two-sided bound (4.17) yields
max ( i , j ) ∈ C ∣ d i j ∣ ≤ C ( m + log D + log ( 1 / t min ) + 1 ) . (92) \max_{(i,j)\in C}|d_{ij}|\le C\left(m+\log D+\log(1/t_{\min})+1\right). \tag*{(92)} ( i , j ) ∈ C max ∣ d ij ∣ ≤ C ( m + log D + log ( 1/ t m i n ) + 1 ) . (92) Apply the constructive linear right inverse in Lemma 2.5 to the approximated data. Its operator bound is C m Cm C m Q ( 3 ) \mathbb{Q}(\sqrt{3}) Q ( 3 ) D p i + s i Dp_i+s_i D p i + s i τ / 8 \tau/8 τ /8 s i = X i − D p i s_i=X_i-Dp_i s i = X i − D p i s i s_i s i u i j u_{ij} u ij
∣ u i j ⋅ ( s j − s i ) − d i j ∣ ≤ τ . (93) \left|u_{ij}\cdot(s_j-s_i)-d_{ij}\right|\le\tau. \tag*{(93)} ∣ u ij ⋅ ( s j − s i ) − d ij ∣ ≤ τ . (93) The right-inverse bound and (92) imply (89) . For example, the 3 \sqrt{3} 3 O ( τ / [ D + poly ( m ) max ( 1 , max ∣ d i j ∣ ) ] ) O(\tau/[D+\operatorname{poly}(m)\max(1,\max|d_{ij}|)]) O ( τ / [ D + poly ( m ) max ( 1 , max ∣ d ij ∣ )]) 2 D − 2 S \sqrt{2}D-2S 2 D − 2 S
There is no requirement to realize prescribed distances exactly. For a contact with δ = s j − s i \delta=s_j-s_i δ = s j − s i
∣ D u i j + δ ∣ = D + u i j ⋅ δ + O ( S 2 / D ) . |Du_{ij}+\delta|=D+u_{ij}\cdot\delta+O(S^2/D). ∣ D u ij + δ ∣ = D + u ij ⋅ δ + O ( S 2 / D ) . Actual coordinates automatically satisfy all vector-cycle identities; the displayed per-edge correction is already included in Lemma 4.5 and is not accumulated through a further nonlinear solve. Put e i j = u i j ⋅ ( s j − s i ) − d i j e_{ij}=u_{ij}\cdot(s_j-s_i)-d_{ij} e ij = u ij ⋅ ( s j − s i ) − d ij
Z ∗ T i j ( X 0 ) e − u i j ⋅ ( s j − s i ) = t i j e − e i j , ∣ e i j ∣ ≤ τ . (94) Z_*T_{ij}(X^0)e^{-u_{ij}\cdot(s_j-s_i)}=t_{ij}e^{-e_{ij}}, \qquad|e_{ij}|\le\tau. \tag*{(94)} Z ∗ T ij ( X 0 ) e − u ij ⋅ ( s j − s i ) = t ij e − e ij , ∣ e ij ∣ ≤ τ . (94) Lemma 4.5 therefore gives Z ∗ T i j ( X ) = t i j ( 1 + O ( ε d i s p + τ ) ) Z_*T_{ij}(X)=t_{ij}(1+O(\varepsilon_{\mathrm{disp}}+\tau)) Z ∗ T ij ( X ) = t ij ( 1 + O ( ε disp + τ )) O ( ε r e l ) O(\varepsilon_{\mathrm{rel}}) O ( ε rel ) (90) . In particular no factor e C m e^{Cm} e C m e ∣ d i j ∣ e^{|d_{ij}|} e ∣ d ij ∣ t i j ≤ 1 t_{ij}\le1 t ij ≤ 1 C E c a l C\mathcal{E}_{\mathrm{cal}} C E cal
All computations of X X X W i ∗ W_i^* W i ∗ D D D ρ i \rho_i ρ i
Finally Lemma 4.7 and Z ∗ ≤ 2 e D Z_*\le2e^D Z ∗ ≤ 2 e D Z ∗ ∣ B i j − B j i ∣ ≤ 4 e − 0.86 D Z_*|B_{ij}-B_{ji}|\le4e^{-0.86D} Z ∗ ∣ B ij − B j i ∣ ≤ 4 e − 0.86 D − 1 / 2 -1/2 − 1/2 λ i \lambda_i λ i
If the contact set is empty, the procedure simply rounds D p i Dp_i D p i (89) is S ≤ poly ( n ) ( 1 + log D ) S \le\operatorname{poly}(n)(1+\log D) S ≤ poly ( n ) ( 1 + log D ) (88) , together with m + S ≤ D 1 / 4 m+S \le D^{1/4} m + S ≤ D 1/4
Excluding lower-energy continuum states
We now compare the spectral infimum of the full Coulomb operator with the half-filled Hubbard energy. In particular, the comparison below includes a lower bound on every state outside the selected orbital space.
The proof has three distinct outputs. We first identify the one-particle matrix in orthonormal site modes and prove a fixed positive gap on its entire orthogonal complement. We then bound the many-electron coupling to that complement and eliminate it by a quadratic-form square completion. Only after this all-state lower bound is available do we compare the compressed Coulomb interaction with the Hubbard onsite term. Keeping these steps separate prevents a variational upper bound from being used as a NO-instance lower bound.
We use the rational centers and tuned parameters constructed in Sections 3 and 4. Increasing the fixed polynomial choice of D D D
m + S ≤ D 1 / 4 , D ≥ D 0 , m+S \le D^{1/4}, \qquad D \ge D_0, m + S ≤ D 1/4 , D ≥ D 0 , where the constant D 0 D_0 D 0
∥ ψ i − ϕ i ∥ H 2 ≤ C m / D , ∥ ψ i ∥ H 2 ≤ C , (95) \lVert\psi_i-\phi_i\rVert_{H^2} \le Cm/D, \qquad\lVert\psi_i\rVert_{H^2} \le C, \tag*{(95)} ∥ ψ i − ϕ i ∥ H 2 ≤ C m / D , ∥ ψ i ∥ H 2 ≤ C , (95) ∣ λ i + 1 / 2 ∣ ≤ e − 1.01 D , ∣ ψ i ( y ) ∣ ≤ C e − 0.99 ( r i − 0.01 D ) . (96) \lvert\lambda_i+1/2\rvert\le e^{-1.01D}, \qquad\lvert\psi_i(y)\rvert\le C e^{-0.99(r_i-0.01D)}. \tag*{(96)} ∣ λ i + 1/2 ∣ ≤ e − 1.01 D , ∣ ψ i ( y )∣ ≤ C e − 0.99 ( r i − 0.01 D ) . (96) There is also a constant g w > 0 g_w>0 g w > 0
h i − λ i ≥ g w ( I − ∣ ψ i ⟩ ⟨ ψ i ∣ ) . (97) h_i-\lambda_i \ge g_w\left(I-\lvert\psi_i\rangle\langle\psi_i\rvert\right). \tag*{(97)} h i − λ i ≥ g w ( I − ∣ ψ i ⟩ ⟨ ψ i ∣ ) . (97) All operator inequalities involving an unbounded operator are understood as quadratic-form inequalities. By Lemma 2.4 and the displacement bound,
∣ X i − X j ∣ ≥ D − 2 S ( i ≠ j ) , ∣ X i − X j ∣ ≥ 2 D − 2 S ( { i , j } ∉ C ) . (98) \lvert X_i-X_j\rvert\ge D-2S \quad(i\ne j), \qquad\lvert X_i-X_j\rvert\ge\sqrt{2}D-2S \quad(\{i,j\}\notin C). \tag*{(98)} ∣ X i − X j ∣ ≥ D − 2 S ( i = j ) , ∣ X i − X j ∣ ≥ 2 D − 2 S ({ i , j } ∈ / C ) . (98) The second inequality concerns distinct noncontact sites.
The one-particle matrix
Recall the hole potentials from Section 4, and write
A = h + 1 2 , V i h o l e = ( 1 − χ i ) V i , ϵ i = λ i + 1 2 . A=h+\frac{1}{2}, \qquad V_i^{\mathrm{hole}}=(1-\chi_i)V_i, \qquad\epsilon_i=\lambda_i+\frac{1}{2}. A = h + 2 1 , V i hole = ( 1 − χ i ) V i , ϵ i = λ i + 2 1 . For the chosen orientation i → j i\to j i → j
B i j = ⟨ ψ j , V j h o l e ψ i ⟩ , ϵ h o p = max i → j ∣ Z ∗ B i j − t i j ∣ . B_{ij}=\langle\psi_j,V_j^{\mathrm{hole}}\psi_i\rangle, \qquad\epsilon_{\mathrm{hop}}=\max_{i\to j}\lvert Z_*B_{ij}-t_{ij}\rvert. B ij = ⟨ ψ j , V j hole ψ i ⟩ , ϵ hop = i → j max ∣ Z ∗ B ij − t ij ∣ . The maximum is zero if there are no contacts. Proposition 4.8 gives
ϵ h o p ≤ C E c a l , (99) \epsilon_{\mathrm{hop}} \le C\mathcal{E}_{\mathrm{cal}}, \tag*{(99)} ϵ hop ≤ C E cal , (99) because all the target hoppings are at most one. Let T \mathrm{T} T m m m m m m − t i j -t_{ij} − t ij
Define U : C m → L 2 ( R 3 ) \mathcal{U}:\mathbb{C}^{m}\to L^{2}(\mathbb{R}^{3}) U : C m → L 2 ( R 3 ) U e i = ψ i \mathcal{U}e_i=\psi_i U e i = ψ i G = U ∗ U G=\mathcal{U}^{*}\mathcal{U} G = U ∗ U U \mathcal{U} U U = 5 / 8 U=5/8 U = 5/8
Lemma 5.1 (Residuals and the orthonormalized matrix). For the preceding parameters and sufficiently large D D D
∥ A ψ i ∥ 2 ≤ e − .75 D , ∥ G − I ∥ ≤ e − .85 D . (100) \lVert A\psi_i\rVert_2\le e^{-.75D},\qquad\lVert G-I\rVert\le e^{-.85D}. \tag*{(100)} ∥ A ψ i ∥ 2 ≤ e − .75 D , ∥ G − I ∥ ≤ e − .85 D . (100) In particular G G G The isometry J 0 = U G − 1 / 2 J_0=\mathcal{U}G^{-1/2} J 0 = U G − 1/2 ω i = J 0 e i \omega_i=J_0e_i ω i = J 0 e i
∥ Z ∗ J 0 ∗ A J 0 − T ∥ ≤ δ 1 , (101) \lVert Z_*J_0^*AJ_0-\mathrm{T}\rVert\le\delta_1, \tag*{(101)} ∥ Z ∗ J 0 ∗ A J 0 − T ∥ ≤ δ 1 , (101) δ 1 : = m ϵ h o p + Z ∗ e − 1.01 D + C m Z ∗ e − 1.1 D + C m Z ∗ e − 1.6 D . \delta_1:=m\epsilon_{\mathrm{hop}}+Z_*e^{-1.01D}+CmZ_*e^{-1.1D}+CmZ_*e^{-1.6D}. δ 1 := m ϵ hop + Z ∗ e − 1.01 D + C m Z ∗ e − 1.1 D + C m Z ∗ e − 1.6 D . If P 0 = J 0 J 0 ∗ P_0=J_0J_0^* P 0 = J 0 J 0 ∗ then
∥ A P 0 ∥ ≤ C m e − .75 D , ∥ ω i − ϕ i ∥ H 1 ≤ C m / D . (102) \lVert AP_0\rVert\le C\sqrt{m}\,e^{-.75D},\qquad\lVert\omega_i-\phi_i\rVert_{H^1}\le Cm/D. \tag*{(102)} ∥ A P 0 ∥ ≤ C m e − .75 D , ∥ ω i − ϕ i ∥ H 1 ≤ C m / D . (102) Proof. Since h = h i − ∑ k ≠ i V k h o l e h=h_i-\sum_{k\ne i}V_k^{\mathrm{hole}} h = h i − ∑ k = i V k hole
A ψ i = ϵ i ψ i − ∑ k ≠ i V k h o l e ψ i . (103) A\psi_i=\epsilon_i\psi_i-\sum_{k\ne i}V_k^{\mathrm{hole}}\psi_i. \tag*{(103)} A ψ i = ϵ i ψ i − k = i ∑ V k hole ψ i . (103) The k k k r k ≤ D / 8 r_k\le D/8 r k ≤ D /8
r i ≥ 7 D / 8 − 2 S ( k ≠ i ) . r_i\ge7D/8-2S\qquad(k\ne i). r i ≥ 7 D /8 − 2 S ( k = i ) . Both Coulomb poles of V k V_k V k D D D q , q ′ q,q' q , q ′
∥ V k h o l e ∥ 2 ≤ C D 1 / 2 , ∥ V k h o l e ∥ 1 ≤ C D 2 . (104) \lVert V_k^{\mathrm{hole}}\rVert_2\le CD^{1/2},\qquad\lVert V_k^{\mathrm{hole}}\rVert_1\le CD^2. \tag*{(104)} ∥ V k hole ∥ 2 ≤ C D 1/2 , ∥ V k hole ∥ 1 ≤ C D 2 . (104) Consequently the sum in (103) has norm at most
C m D 1 / 2 exp ( − .85635 D + 1.98 S ) . CmD^{1/2}\exp(-.85635D+1.98S). C m D 1/2 exp ( − .85635 D + 1.98 S ) . Together with the tuning error, this is at most e − .75 D e^{-.75D} e − .75 D D D D
For i ≠ j i\ne j i = j ℓ i j = ∣ X i − X j ∣ \ell_{ij}=\lvert X_i-X_j\rvert ℓ ij = ∣ X i − X j ∣
C exp ( − .99 ( r i + r j ) + .0198 D ) . C\exp(-.99(r_i+r_j)+.0198D). C exp ( − .99 ( r i + r j ) + .0198 D ) . Use r i + r j ≥ ℓ i j r_i+r_j\ge\ell_{ij} r i + r j ≥ ℓ ij .90 .90 .90 ∫ e − .09 ( r i + r j ) d y ≤ ∫ e − .09 r i d y ≤ C \int e^{-.09(r_i+r_j)}\,dy\le\int e^{-.09r_i}\,dy\le C ∫ e − .09 ( r i + r j ) d y ≤ ∫ e − .09 r i d y ≤ C
∣ G i j ∣ ≤ C e − .90 ℓ i j + .0198 D ≤ C e − .8802 D + 1.8 S . \lvert G_{ij}\rvert\le Ce^{-.90\ell_{ij}+.0198D}\le Ce^{-.8802D+1.8S}. ∣ G ij ∣ ≤ C e − .90 ℓ ij + .0198 D ≤ C e − .8802 D + 1.8 S . The diagonal entries are one. The row-sum estimate and m + S ≤ D 1 / 4 m+S \le D^{1/4} m + S ≤ D 1/4 (100) .
Let M = U ∗ A U M=U^*AU M = U ∗ A U k ≠ i , j k\ne i,j k = i , j r i r_i r i r j r_j r j 7 D / 8 − 2 S 7D/8-2S 7 D /8 − 2 S k k k (96) and (104) therefore imply
∑ k ≠ i , j ∣ ⟨ ψ j , V k h o l e ψ i ⟩ ∣ ≤ C m D 2 e − 1.7127 D + 3.96 S . \sum_{k\ne i,j} \lvert\langle\psi_j,V_k^{\mathrm{hole}}\psi_i\rangle\rvert\le CmD^2 e^{-1.7127D+3.96S}. k = i , j ∑ ∣⟨ ψ j , V k hole ψ i ⟩∣ ≤ C m D 2 e − 1.7127 D + 3.96 S . The same estimate bounds ∑ k ≠ i ⟨ ψ i , V k h o l e ψ i ⟩ \sum_{k\ne i}\langle\psi_i,V_k^{\mathrm{hole}}\psi_i\rangle ∑ k = i ⟨ ψ i , V k hole ψ i ⟩ j j j r i + r j ≥ 2 D − 2 S r_i+r_j\ge\sqrt{2}D-2S r i + r j ≥ 2 D − 2 S
∣ ⟨ ψ j , V j h o l e ψ i ⟩ ∣ ≤ C D 2 e − ( .99 2 − .0198 ) D + 1.98 S . \lvert\langle\psi_j,V_j^{\mathrm{hole}}\psi_i\rangle\rvert\le CD^2e^{-(.99\sqrt{2}-.0198)D+1.98S}. ∣⟨ ψ j , V j hole ψ i ⟩∣ ≤ C D 2 e − ( .99 2 − .0198 ) D + 1.98 S . Here .99 2 − .0198 > 1.38 .99\sqrt{2}-.0198>1.38 .99 2 − .0198 > 1.38 e − 1.1 D e^{-1.1D} e − 1.1 D i ≠ j i\ne j i = j ϵ i G j i \epsilon_iG_{ji} ϵ i G j i e − 1.86 D e^{-1.86D} e − 1.86 D
∣ M i i − ϵ i ∣ ≤ C e − 1.1 D , ∣ M j i + B i j ∣ ≤ C e − 1.1 D ( i → j a chosen contact ) , ∣ M i j ∣ ≤ C e − 1.1 D ( i ≠ j , { i , j } ∉ C ) . (105) \begin{aligned}
\lvert M_{ii}-\epsilon_i\rvert&\le Ce^{-1.1D},\\
\lvert M_{ji}+B_{ij}\rvert&\le Ce^{-1.1D} \qquad(i\to j\text{ a chosen contact}),\\
\lvert M_{ij}\rvert&\le Ce^{-1.1D} \qquad(i\ne j,\{i,j\}\notin\mathcal{C}). \tag*{(105)}
\end{aligned} ∣ M ii − ϵ i ∣ ∣ M j i + B ij ∣ ∣ M ij ∣ ≤ C e − 1.1 D , ≤ C e − 1.1 D ( i → j a chosen contact ) , ≤ C e − 1.1 D ( i = j , { i , j } ∈ / C ) . (105) The exactly symmetric matrix M M M e − 1.01 D e^{-1.01D} e − 1.01 D m e − 1.01 D me^{-1.01D} m e − 1.01 D T T T
∥ Z ∗ M − T ∥ ≤ m ϵ h o p + Z ∗ e − 1.01 D + C m Z ∗ e − 1.1 D . \lVert Z_*M-T\rVert\le m\epsilon_{\mathrm{hop}}+Z_*e^{-1.01D}+CmZ_*e^{-1.1D}. ∥ Z ∗ M − T ∥ ≤ m ϵ hop + Z ∗ e − 1.01 D + C m Z ∗ e − 1.1 D . For completeness, the residual estimate controls the entire matrix, without using a dimension-dependent bound on a many-body space:
∥ A U ∥ ≤ m e − .75 D , ∥ U ∥ = ∥ G ∥ 1 / 2 ≤ 2 , ∥ M ∥ ≤ C m e − .75 D . \lVert AU\rVert\le\sqrt{m}\,e^{-.75D},\qquad\lVert U\rVert=\lVert G\rVert^{1/2}\le2,\qquad\lVert M\rVert\le C\sqrt{m}\,e^{-.75D}. ∥ A U ∥ ≤ m e − .75 D , ∥ U ∥ = ∥ G ∥ 1/2 ≤ 2 , ∥ M ∥ ≤ C m e − .75 D . Functional calculus on the positive matrix G G G ∥ G − 1 / 2 − I ∥ ≤ C e − .85 D \lVert G^{-1/2}-I\rVert\le Ce^{-.85D} ∥ G − 1/2 − I ∥ ≤ C e − .85 D ∥ G − 1 / 2 ∥ ≤ 2 \lVert G^{-1/2}\rVert\le2 ∥ G − 1/2 ∥ ≤ 2
∥ G − 1 / 2 M G − 1 / 2 − M ∥ ≤ C m e − 1.6 D ≤ C m e − 1.6 D . (106) \lVert G^{-1/2}MG^{-1/2}-M\rVert\le C\sqrt{m}\,e^{-1.6D}\le Cme^{-1.6D}. \tag*{(106)} ∥ G − 1/2 M G − 1/2 − M ∥ ≤ C m e − 1.6 D ≤ C m e − 1.6 D . (106) This orthonormalizes the matrix of the shifted operator h + 1 / 2 h+1/2 h + 1/2 M = U ∗ h U + 1 2 G M=U^*hU+\frac{1}{2}G M = U ∗ h U + 2 1 G 1 2 G \frac{1}{2}G 2 1 G 1 2 I \frac{1}{2}I 2 1 I (101) follows.
The same residual estimate applied to A J 0 = A U G − 1 / 2 AJ_0=AUG^{-1/2} A J 0 = A U G − 1/2 (102) . Finally, as a map into H 1 H^1 H 1 U U U C m C\sqrt{m} C m (95) . Thus
∥ ω i − ψ i ∥ H 1 ≤ C m e − .85 D . \lVert\omega_i-\psi_i\rVert_{H^1}\le C\sqrt{m}\,e^{-.85D}. ∥ ω i − ψ i ∥ H 1 ≤ C m e − .85 D . Combining this with ∥ ψ i − ϕ i ∥ H 1 ≤ C m / D \lVert\psi_i-\phi_i\rVert_{H^1}\le Cm/D ∥ ψ i − ϕ i ∥ H 1 ≤ C m / D
We now include spin. Set
h = L 2 ( R 3 ; C 2 ) , P = P 0 ⊗ I 2 , J = J 0 ⊗ I 2 . \mathfrak{h}=L^2(\mathbb{R}^3;\mathbb{C}^2),\qquad P=P_0\otimes I_2,\qquad\mathcal{J}=\mathcal{J}_0\otimes I_2. h = L 2 ( R 3 ; C 2 ) , P = P 0 ⊗ I 2 , J = J 0 ⊗ I 2 . Thus P 0 P_0 P 0 m m m P P P 2 m 2m 2 m A A A
A gap on the entire one-particle complement
Spatial localization separates the nuclei from the exterior, where the attraction is uniformly weak. We use the IMS product-rule identity (compare [14 ] , Lemma 3.1); its kinetic-error coefficient is 1 / 2 1/2 1/2 − Δ / 2 -\Delta/2 − Δ/2
Proposition 5.2 (Continuum complement gap). There is a constant g > 0 g > 0 g > 0 D D D
( I − P ) A ( I − P ) ≥ g ( I − P ) . (I-P)A(I-P) \ge g(I-P). ( I − P ) A ( I − P ) ≥ g ( I − P ) . The inequality holds on the full one-particle form domain, including functions supported between the nuclei or at infinity.
Proof. Choose smooth real functions θ 0 , θ 1 , … , θ m \theta_0,\theta_1,\ldots,\theta_m θ 0 , θ 1 , … , θ m
∑ a = 0 m θ a 2 = 1 , θ i = 1 on r i ≤ D / 5 , supp θ i ⊆ { r i ≤ D / 4 } . \sum_{a=0}^{m}\theta_a^2 = 1,\qquad\theta_i = 1\ \text{on } r_i \le D/5,\qquad\operatorname{supp}\theta_i \subseteq\{r_i \le D/4\}. a = 0 ∑ m θ a 2 = 1 , θ i = 1 on r i ≤ D /5 , supp θ i ⊆ { r i ≤ D /4 } . The balls of radius D / 4 D/4 D /4 (98) . In the annulus D / 5 < r i < D / 4 D/5 < r_i < D/4 D /5 < r i < D /4 θ i = cos f ( r i / D ) \theta_i = \cos f(r_i/D) θ i = cos f ( r i / D ) θ 0 = sin f ( r i / D ) \theta_0 = \sin f(r_i/D) θ 0 = sin f ( r i / D ) f f f π / 2 \pi/2 π /2 θ 0 = 0 \theta_0 = 0 θ 0 = 0
∑ a = 0 m ∣ ∇ θ a ∣ 2 ≤ C D − 2 . (107) \sum_{a=0}^{m} \lvert\nabla\theta_a\rvert^2 \le CD^{-2}. \tag*{(107)} a = 0 ∑ m ∣ ∇ θ a ∣ 2 ≤ C D − 2 . (107) This construction also verifies smoothness at the boundaries.
Let u ∈ H 1 ( R 3 ; C 2 ) u \in H^1(\mathbb{R}^3;\mathbb{C}^2) u ∈ H 1 ( R 3 ; C 2 ) P u = 0 Pu = 0 P u = 0 ∥ u ∥ = 1 \lVert u\rVert= 1 ∥ u ∥ = 1 σ \sigma σ
b i σ : = ⟨ ψ i , θ i u σ ⟩ = ⟨ ( θ i − 1 ) ψ i , u σ ⟩ . b_{i\sigma} := \langle\psi_i,\theta_i u_\sigma\rangle= \langle(\theta_i-1)\psi_i,u_\sigma\rangle. b iσ := ⟨ ψ i , θ i u σ ⟩ = ⟨( θ i − 1 ) ψ i , u σ ⟩ . The tail estimate, integrated over r i ≥ D / 5 r_i \ge D/5 r i ≥ D /5
∥ ( θ i − 1 ) ψ i ∥ ≤ C poly ( D ) e − .99 ( .2 − .01 ) D ≤ e − .18 D . \lVert(\theta_i-1)\psi_i\rVert\le C\operatorname{poly}(D)e^{-.99(.2-.01)D} \le e^{-.18D}. ∥( θ i − 1 ) ψ i ∥ ≤ C poly ( D ) e − .99 ( .2 − .01 ) D ≤ e − .18 D . Here the polynomial comes only from radial integration and has fixed degree. Consequently
∑ i , σ ∣ b i σ ∣ 2 ≤ m e − .36 D . (108) \sum_{i,\sigma}\lvert b_{i\sigma}\rvert^2 \le me^{-.36D}. \tag*{(108)} i , σ ∑ ∣ b iσ ∣ 2 ≤ m e − .36 D . (108) There is no assumption that a localized vector is exactly orthogonal to the local ground state.
On supp θ i \operatorname{supp}\theta_i supp θ i k ≠ i k \ne i k = i
r k ≥ D − 2 S − D / 4 = 3 D / 4 − 2 S > D / 8. r_k \ge D-2S-D/4 = 3D/4-2S > D/8. r k ≥ D − 2 S − D /4 = 3 D /4 − 2 S > D /8. Hence χ k = 1 \chi_k = 1 χ k = 1 h = h i h = h_i h = h i 0 < g l o c ≤ min ( g w / 2 , 1 / 4 ) 0 < g_{\mathrm{loc}} \le\min(g_w/2,1/4) 0 < g loc ≤ min ( g w /2 , 1/4 ) (97) and (96) , for large D D D
⟨ θ i u , A θ i u ⟩ ≥ g l o c ∥ θ i u ∥ 2 − ( g l o c + e − 1.01 D ) ∑ σ ∣ b i σ ∣ 2 . \langle\theta_i u,A\theta_i u\rangle\ge g_{\mathrm{loc}}\lVert\theta_i u\rVert^2-\left(g_{\mathrm{loc}}+e^{-1.01D}\right)\sum_{\sigma}\lvert b_{i\sigma}\rvert^2. ⟨ θ i u , A θ i u ⟩ ≥ g loc ∥ θ i u ∥ 2 − ( g loc + e − 1.01 D ) σ ∑ ∣ b iσ ∣ 2 . On supp θ 0 \operatorname{supp}\theta_0 supp θ 0 D / 5 D/5 D /5 D / 5 − ρ i ≥ D / 5 − 4 ≥ D / 6 D/5-\rho_i \ge D/5-4 \ge D/6 D /5 − ρ i ≥ D /5 − 4 ≥ D /6 q + q ′ ≤ 2 q+q' \le2 q + q ′ ≤ 2 D D D C m / D C_m/D C m / D
⟨ θ 0 u , A θ 0 u ⟩ ≥ ( 1 / 2 − C m / D ) ∥ θ 0 u ∥ 2 ≥ g l o c ∥ θ 0 u ∥ 2 . \langle\theta_0u,A\theta_0u\rangle\ge(1/2-C_m/D)\lVert\theta_0u\rVert^2 \ge g_{\mathrm{loc}}\lVert\theta_0u\rVert^2. ⟨ θ 0 u , A θ 0 u ⟩ ≥ ( 1/2 − C m / D ) ∥ θ 0 u ∥ 2 ≥ g loc ∥ θ 0 u ∥ 2 . The kinetic product rule yields the IMS identity
⟨ u , A u ⟩ = ∑ a = 0 m ⟨ θ a u , A θ a u ⟩ − 1 2 ∫ ∑ a = 0 m ∣ ∇ θ a ∣ 2 ∣ u ∣ 2 . \langle u,Au\rangle=\sum_{a=0}^{m}\langle\theta_a u,A\theta_a u\rangle-\frac{1}{2}\int\sum_{a=0}^{m}\lvert\nabla\theta_a\rvert^2\lvert u\rvert^2. ⟨ u , A u ⟩ = a = 0 ∑ m ⟨ θ a u , A θ a u ⟩ − 2 1 ∫ a = 0 ∑ m ∣ ∇ θ a ∣ 2 ∣ u ∣ 2 . It holds first for smooth functions and then on H 1 H^1 H 1 (107) and (108) gives
⟨ u , A u ⟩ ≥ g l o c − C m e − .36 D − C D − 2 ≥ g l o c / 2. \langle u,Au\rangle\ge g_{\mathrm{loc}}-C_m e^{-.36D}-CD^{-2}\ge g_{\mathrm{loc}}/2. ⟨ u , A u ⟩ ≥ g loc − C m e − .36 D − C D − 2 ≥ g loc /2. Taking g = g l o c / 2 g=g_{\mathrm{loc}}/2 g = g loc /2
Let
H m = ⋀ m h , Π = ⋀ m P , Q m = I − Π . \mathcal{H}_m=\bigwedge^m\mathfrak{h},\qquad\Pi=\bigwedge^m P,\qquad Q_m=I-\Pi. H m = ⋀ m h , Π = ⋀ m P , Q m = I − Π. Here Π \Pi Π m m m 2 m 2m 2 m d Γ ( B ) = ∑ ℓ = 1 m B ( ℓ ) \mathrm{d}\Gamma(B)=\sum_{\ell=1}^{m}B^{(\ell)} d Γ ( B ) = ∑ ℓ = 1 m B ( ℓ ) W e e = ∑ 1 ≤ ℓ < k ≤ m ∣ y ℓ − y k ∣ − 1 W_{\mathrm{ee}}=\sum_{1\le\ell<k\le m}\lvert y_\ell-y_k\rvert^{-1} W ee = ∑ 1 ≤ ℓ < k ≤ m ∣ y ℓ − y k ∣ − 1
Under the unitary dilation y ℓ = Z ∗ x ℓ y_\ell=Z_*x_\ell y ℓ = Z ∗ x ℓ
H Z ∗ 2 = d Γ ( h ) + W e e Z ∗ . (109) \frac{H}{Z_*^2}=\mathrm{d}\Gamma(h)+\frac{W_{\mathrm{ee}}}{Z_*}. \tag*{(109)} Z ∗ 2 H = d Γ ( h ) + Z ∗ W ee . (109) We henceforth use this unitary identification and set
H = H Z ∗ 2 + m 2 = d Γ ( A ) + W e e Z ∗ . (110) \mathcal{H}=\frac{H}{Z_*^2}+\frac{m}{2}=\mathrm{d}\Gamma(A)+\frac{W_{\mathrm{ee}}}{Z_*}. \tag*{(110)} H = Z ∗ 2 H + 2 m = d Γ ( A ) + Z ∗ W ee . (110) To compare its spectral infimum with its compression to Π \Pi Π Q m Q_m Q m Q m H Π Q_m\mathcal{H}\Pi Q m H Π d Γ ( A ) \mathrm{d}\Gamma(A) d Γ ( A ) Ran Π \operatorname{Ran}\Pi Ran Π
We briefly specify the domains used below. On the antisymmetric subspace, the kinetic form domain is H 1 H^1 H 1 3 m 3m 3 m H 2 H^2 H 2 H 1 H^1 H 1 L 2 L^2 L 2 ∥ V u ∥ 2 ≤ C i n s t ∥ u ∥ H 1 \lVert Vu\rVert_2\le C_{\mathrm{inst}}\lVert u\rVert_{H^1} ∥ V u ∥ 2 ≤ C inst ∥ u ∥ H 1 ∥ ∇ u ∥ 2 ≤ ε ∥ Δ u ∥ 2 + C ε ∥ u ∥ 2 \lVert\nabla u\rVert_2\le\varepsilon\lVert\Delta u\rVert_2+C_\varepsilon\lVert u\rVert_2 ∥ ∇ u ∥ 2 ≤ ε ∥ Δ u ∥ 2 + C ε ∥ u ∥ 2 H 2 H^2 H 2 H 1 H^1 H 1 ω i \omega_i ω i H 2 ( R 3 ) H^2(\mathbb{R}^3) H 2 ( R 3 ) Ran Π \operatorname{Ran}\Pi Ran Π Π \Pi Π Q m Q_m Q m
Lemma 5.3 (A dimension-independent fermionic Hardy bound). Uniformly over all normalized u ∈ Ran Π u \in\operatorname{Ran}\Pi u ∈ Ran Π
∑ ℓ = 1 m ∥ ∇ y ℓ u ∥ 2 2 ≤ C m , ∥ W e e Π ∥ ≤ C m 2 . (111) \sum_{\ell=1}^{m}\|\nabla_{y_\ell}u\|_2^2 \le Cm,\qquad\|W_{\mathrm{ee}}\Pi\| \le Cm^2. \tag*{(111)} ℓ = 1 ∑ m ∥ ∇ y ℓ u ∥ 2 2 ≤ C m , ∥ W ee Π∥ ≤ C m 2 . (111) The constants do not depend on the dimension ( 2 m m ) \binom{2m}{m} ( m 2 m ) of Ran Π \operatorname{Ran}\Pi Ran Π
Proof. Let ( K r a w ) i j = ⟨ ∇ ψ i , ∇ ψ j ⟩ (K_{\mathrm{raw}})_{ij}=\langle\nabla\psi_i,\nabla\psi_j\rangle ( K raw ) ij = ⟨ ∇ ψ i , ∇ ψ j ⟩ C m Cm C m
K 0 = G − 1 / 2 K r a w G − 1 / 2 . K_0=G^{-1/2}K_{\mathrm{raw}}G^{-1/2}. K 0 = G − 1/2 K raw G − 1/2 . Hence
Tr K 0 = Tr ( K r a w G − 1 ) ≤ ∥ G − 1 ∥ Tr K r a w ≤ C m . \operatorname{Tr}K_0=\operatorname{Tr}(K_{\mathrm{raw}}G^{-1})\le\|G^{-1}\|\operatorname{Tr}K_{\mathrm{raw}}\le Cm. Tr K 0 = Tr ( K raw G − 1 ) ≤ ∥ G − 1 ∥ Tr K raw ≤ C m . The spin matrix K = K 0 ⊗ I 2 K=K_0\otimes I_2 K = K 0 ⊗ I 2
For any normalized fermionic u u u γ \gamma γ 0 ≤ γ ≤ I 0\le\gamma\le I 0 ≤ γ ≤ I v v v γ \gamma γ ∥ c ( v ) u ∥ 2 \|c(v)u\|^2 ∥ c ( v ) u ∥ 2 c ( v ) c ( v ) † + c ( v ) † c ( v ) = ∥ v ∥ 2 I c(v)c(v)^\dagger+c(v)^\dagger c(v)=\|v\|^2I c ( v ) c ( v ) † + c ( v ) † c ( v ) = ∥ v ∥ 2 I ∥ v ∥ 2 \|v\|^2 ∥ v ∥ 2
∑ ℓ ∥ ∇ y ℓ u ∥ 2 2 = Tr ( γ K ) ≤ Tr K ≤ C m . \sum_{\ell}\|\nabla_{y_\ell}u\|_2^2=\operatorname{Tr}(\gamma K)\le\operatorname{Tr}K\le Cm. ℓ ∑ ∥ ∇ y ℓ u ∥ 2 2 = Tr ( γ K ) ≤ Tr K ≤ C m . For a pair ℓ < k \ell<k ℓ < k y ℓ y_\ell y ℓ y k y_k y k
∥ ∣ y ℓ − y k ∣ − 1 u ∥ 2 2 ≤ 2 ∥ ∇ y ℓ u ∥ 2 2 . \||y_\ell-y_k|^{-1}u\|_2^2\le2\|\nabla_{y_\ell}u\|_2^2. ∥∣ y ℓ − y k ∣ − 1 u ∥ 2 2 ≤ 2∥ ∇ y ℓ u ∥ 2 2 . Integration over the remaining variables and spin components justifies this inequality for the full wavefunction. Put b m = m ( m − 1 ) / 2 b_m=m(m-1)/2 b m = m ( m − 1 ) /2 m ≥ 2 m\ge2 m ≥ 2
∥ W e e u ∥ 2 2 ≤ b m ∑ ℓ < k ∥ ∣ y ℓ − y k ∣ − 1 u ∥ 2 2 ≤ 4 b m ∑ ℓ < k ∥ ∇ y ℓ u ∥ 2 2 ≤ 4 b m ( m − 1 ) ∑ ℓ ∥ ∇ y ℓ u ∥ 2 2 ≤ C m 4 . \begin{aligned}
\|W_{\mathrm{ee}}u\|_2^2
&\le b_m\sum_{\ell<k}\||y_\ell-y_k|^{-1}u\|_2^2\\
&\le4b_m\sum_{\ell<k}\|\nabla_{y_\ell}u\|_2^2\\
&\le4b_m(m-1)\sum_{\ell}\|\nabla_{y_\ell}u\|_2^2\le Cm^4.
\end{aligned} ∥ W ee u ∥ 2 2 ≤ b m ℓ < k ∑ ∥∣ y ℓ − y k ∣ − 1 u ∥ 2 2 ≤ 4 b m ℓ < k ∑ ∥ ∇ y ℓ u ∥ 2 2 ≤ 4 b m ( m − 1 ) ℓ ∑ ∥ ∇ y ℓ u ∥ 2 2 ≤ C m 4 . For m = 1 m=1 m = 1
Lemma 5.4 (Eliminating the many-particle complement). For sufficiently large D D D let E Π = min Spec ( Π H Π ∣ Ran Π ) E_\Pi=\min\operatorname{Spec}(\Pi\mathcal{H}\Pi|_{\operatorname{Ran}\Pi}) E Π = min Spec ( Π H Π ∣ Ran Π ) Then
0 ≤ E Π − inf Spec H ≤ e − 1.2 D . (112) 0\le E_\Pi-\inf\operatorname{Spec}\mathcal{H}\le e^{-1.2D}. \tag*{(112)} 0 ≤ E Π − inf Spec H ≤ e − 1.2 D . (112) More explicitly, the upper bound can be taken to be
C ( m 3 / 2 e − .75 D + m 2 Z ∗ ) 2 . C\left(m^{3/2}e^{-.75D}+\frac{m^2}{Z_*}\right)^2. C ( m 3/2 e − .75 D + Z ∗ m 2 ) 2 . Proof. Let ε = ∥ A P ∥ ≤ C m e − .75 D \varepsilon= \lVert AP\rVert\le C\sqrt{m}e^{-.75D} ε = ∥ A P ∥ ≤ C m e − .75 D
R o f f = P A ( I − P ) + ( I − P ) A P R_{\mathrm{off}} = PA(I-P) + (I-P)AP R off = P A ( I − P ) + ( I − P ) A P extends to a bounded self-adjoint operator, of norm ∥ ( I − P ) A P ∥ ≤ ε \lVert(I-P)AP\rVert\le\varepsilon ∥( I − P ) A P ∥ ≤ ε A A A
A − R o f f = P A P + ( I − P ) A ( I − P ) ≥ − ε P + g ( I − P ) A-R_{\mathrm{off}} = PAP + (I-P)A(I-P) \ge-\varepsilon P + g(I-P) A − R off = P A P + ( I − P ) A ( I − P ) ≥ − εP + g ( I − P ) by Proposition 5.2. Consequently, on the full many-particle form domain,
d Γ ( A ) ≥ g N ⊥ − 2 m ε I , N ⊥ = d Γ ( I − P ) . (113) \mathrm{d}\Gamma(A) \ge gN_{\perp} - 2m\varepsilon I,\qquad N_{\perp} = \mathrm{d}\Gamma(I-P). \tag*{(113)} d Γ ( A ) ≥ g N ⊥ − 2 m ε I , N ⊥ = d Γ ( I − P ) . (113) Here the low diagonal can cost at most m ε m\varepsilon m ε ∥ d Γ ( R o f f ) ∥ ≤ m ε \lVert\mathrm{d}\Gamma(R_{\mathrm{off}})\rVert\le m\varepsilon ∥ d Γ ( R off )∥ ≤ m ε ( I − P ) ( ℓ ) (I-P)^{(\ell)} ( I − P ) ( ℓ ) N ⊥ N_{\perp} N ⊥ Ran Π \operatorname{Ran}\Pi Ran Π N ⊥ ≥ Q m N_{\perp} \ge Q_m N ⊥ ≥ Q m
The repulsion form is nonnegative on every wavefunction, so its compression to Q m Q_m Q m W e e W_{\mathrm{ee}} W ee Π \Pi Π (113) gives
Q m H Q m ≥ ( g − 2 m ε ) Q m ≥ c Q m , c = g / 2 , (114) Q_mHQ_m \ge(g-2m\varepsilon)Q_m \ge cQ_m,\qquad c=g/2, \tag*{(114)} Q m H Q m ≥ ( g − 2 m ε ) Q m ≥ c Q m , c = g /2 , (114) for large D D D
By Lemma 5.3, the coupling from the finite subspace has operator norm at most
β : = ∥ Q m H Π ∥ ≤ m ε + C m 2 Z ∗ ≤ C m 3 / 2 e − .75 D + C m 2 Z ∗ . (115) \beta:= \lVert Q_mH\Pi\rVert\le m\varepsilon+\frac{Cm^2}{Z_*}\le Cm^{3/2}e^{-.75D}+\frac{Cm^2}{Z_*}. \tag*{(115)} β := ∥ Q m H Π ∥ ≤ m ε + Z ∗ C m 2 ≤ C m 3/2 e − .75 D + Z ∗ C m 2 . (115) The first term follows because the block-diagonal one-particle sum preserves Ran Π \operatorname{Ran}\Pi Ran Π
∥ Π H Π ∥ ≤ m ε + C m 2 Z ∗ = o ( 1 ) . \lVert\Pi H\Pi\rVert\le m\varepsilon+\frac{Cm^2}{Z_*}=o(1). ∥ Π H Π ∥ ≤ m ε + Z ∗ C m 2 = o ( 1 ) . For any form-domain vector u = p + v u=p+v u = p + v p = Π u p=\Pi u p = Π u v = Q m u v=Q_m u v = Q m u
⟨ u , H u ⟩ ≥ ⟨ p , H p ⟩ + c ∥ v ∥ 2 − 2 β ∥ p ∥ ∥ v ∥ ≥ ⟨ p , H p ⟩ − 2 β 2 c ∥ p ∥ 2 + c 2 ∥ v ∥ 2 . \begin{aligned}
\langle u,Hu\rangle\ge\langle p,Hp\rangle+c\lVert v\rVert^2-2\beta\lVert p\rVert\lVert v\rVert\\
&\ge\langle p,Hp\rangle-\frac{2\beta^2}{c}\lVert p\rVert^2+\frac{c}{2}\lVert v\rVert^2.
\end{aligned} ⟨ u , H u ⟩ ≥ ⟨ p , H p ⟩ + c ∥ v ∥ 2 − 2 β ∥ p ∥ ∥ v ∥ ≥ ⟨ p , H p ⟩ − c 2 β 2 ∥ p ∥ 2 + 2 c ∥ v ∥ 2 . Equivalently, this is the form inequality
H ≥ Π H Π − 2 β 2 c Π + c 2 Q m . H\ge\Pi H\Pi-\frac{2\beta^2}{c}\Pi+\frac{c}{2}Q_m. H ≥ Π H Π − c 2 β 2 Π + 2 c Q m . Since the finite block has norm o ( 1 ) o(1) o ( 1 ) E Π − 2 β 2 / c E_{\Pi}-2\beta^2/c E Π − 2 β 2 / c D D D Ran Π \operatorname{Ran}\Pi Ran Π Z ∗ ≥ e D / 2 Z_* \ge e^{D/2} Z ∗ ≥ e D /2 e − 1.2 D e^{-1.2D} e − 1.2 D
In particular, the explicit error in this lemma, after multiplication by Z ∗ Z_* Z ∗
C ( m 3 e − .5 D + m 7 / 2 e − .75 D + m 4 e − D ) . (116) C\left(m^3e^{-.5D}+m^{7/2}e^{-.75D}+m^4e^{-D}\right). \tag*{(116)} C ( m 3 e − .5 D + m 7/2 e − .75 D + m 4 e − D ) . (116) The weaker bound Z ∗ e − 1.2 D ≤ 2 e − .2 D Z_*e^{-1.2D}\le2e^{-.2D} Z ∗ e − 1.2 D ≤ 2 e − .2 D
The compressed Coulomb interaction
The full-space energy is now controlled by its finite-mode compression, with an error that remains small after multiplication by Z ∗ Z_* Z ∗ O ( m 2 / D ) O(m^2/D) O ( m 2 / D )
Identify Ran Π \operatorname{Ran}\Pi Ran Π Λ m ( C m ⊗ C 2 ) \Lambda^m(\mathbb{C}^m \otimes\mathbb{C}^2) Λ m ( C m ⊗ C 2 ) J m = Λ m J \mathcal{J}_m=\Lambda^m\mathcal{J} J m = Λ m J
W e f f = J m ∗ W e e J m . W_{\mathrm{eff}}=\mathcal{J}_m^*W_{\mathrm{ee}}\mathcal{J}_m. W eff = J m ∗ W ee J m . Creation and annihilation operators below refer to these orthonormal abstract site modes.
Lemma 5.5 (Coulomb coefficients in the site modes). For sufficiently large D D D
∥ W e f f − U ∑ i = 1 m n i ↑ n i ↓ ∥ ≤ C m 5 D , U = 5 8 . (117) \left\|W_{\mathrm{eff}}-U\sum_{i=1}^{m}n_{i\uparrow}n_{i\downarrow}\right\|\leq\frac{Cm^5}{D},\qquad U=\frac{5}{8}. \tag*{(117)} W eff − U i = 1 ∑ m n i ↑ n i ↓ ≤ D C m 5 , U = 8 5 . (117) Proof. For four spatial functions define
V ( a , b ; c , d ) = ∬ a ( x ) b ( y ) ‾ c ( x ) d ( y ) ∣ x − y ∣ d x d y . \mathcal{V}(a,b;c,d)=\iint\frac{\overline{a(x)b(y)}c(x)d(y)}{\lvert x-y\rvert}\,\mathrm{d}x\,\mathrm{d}y. V ( a , b ; c , d ) = ∬ ∣ x − y ∣ a ( x ) b ( y ) c ( x ) d ( y ) d x d y . Hardy in the x x x
∣ V ( a , b ; c , d ) ∣ ≤ 2 ∥ a ∥ 2 ∥ b ∥ 2 ∥ ∇ c ∥ 2 ∥ d ∥ 2 . (118) \lvert\mathcal{V}(a,b;c,d)\rvert\leq2\|a\|_2\|b\|_2\|\nabla c\|_2\|d\|_2. \tag*{(118)} ∣ V ( a , b ; c , d )∣ ≤ 2∥ a ∥ 2 ∥ b ∥ 2 ∥∇ c ∥ 2 ∥ d ∥ 2 . (118) Telescoping a change of the four arguments therefore makes this coefficient Lipschitz in their H 1 H^1 H 1 H 1 H^1 H 1
The coefficients of W e f f W_{\mathrm{eff}} W eff ω i \omega_i ω i (102) permits their replacement by ϕ i \phi_i ϕ i C m / D Cm/D C m / D H 1 H^1 H 1 1 2 c d † c c † c b c a \frac{1}{2}c_d^\dagger c_c^\dagger c_b c_a 2 1 c d † c c † c b c a ( 2 m ) 4 (2m)^4 ( 2 m ) 4 C m 5 / D Cm^5/D C m 5 / D ϕ i \phi_i ϕ i
Choose a fixed smooth radial cutoff equal to one on r i ≤ D / 5 r_i\leq D/5 r i ≤ D /5 r i ≥ D / 4 r_i\geq D/4 r i ≥ D /4 C / D C/D C / D ζ i \zeta_i ζ i ϕ i \phi_i ϕ i
∥ ζ i − ϕ i ∥ H 1 ≤ C poly ( D ) e − D / 5 ≤ e − .18 D . \|\zeta_i-\phi_i\|_{H^1}\leq C\operatorname{poly}(D)e^{-D/5}\leq e^{-.18D}. ∥ ζ i − ϕ i ∥ H 1 ≤ C poly ( D ) e − D /5 ≤ e − .18 D . The supports of the ζ i \zeta_i ζ i (118) , replacing all ϕ i \phi_i ϕ i ζ i \zeta_i ζ i C ′ m 4 e − .18 D C'm^4e^{-.18D} C ′ m 4 e − .18 D
For the truncated functions a nonzero coefficient must have the same site label for both factors at x x x y y y i ≠ j i\neq j i = j
0 ≤ V i j d i r = ∬ ∣ ζ i ( x ) ∣ 2 ∣ ζ j ( y ) ∣ 2 ∣ x − y ∣ d x d y ≤ 1 D / 2 − 2 S ≤ C D . 0\leq V_{ij}^{\mathrm{dir}}=\iint\frac{\lvert\zeta_i(x)\rvert^2\lvert\zeta_j(y)\rvert^2}{\lvert x-y\rvert}\,\mathrm{d}x\,\mathrm{d}y\leq\frac{1}{D/2-2S}\leq\frac{C}{D}. 0 ≤ V ij dir = ∬ ∣ x − y ∣ ∣ ζ i ( x ) ∣ 2 ∣ ζ j ( y ) ∣ 2 d x d y ≤ D /2 − 2 S 1 ≤ D C . using ∥ ζ i ∥ 2 ≤ 1 \lVert\zeta_i\rVert_2 \le1 ∥ ζ i ∥ 2 ≤ 1 ∑ i < j V i j d i r n i n j \sum_{i<j} V_{ij}^{\mathrm{dir}} n_i n_j ∑ i < j V ij dir n i n j n i = n i ↑ + n i ↓ n_i = n_{i\uparrow} + n_{i\downarrow} n i = n i ↑ + n i ↓ C m 2 / D Cm^2/D C m 2 / D
The onsite spatial integral differs from V ( ϕ i , ϕ i ; ϕ i , ϕ i ) = U \mathcal{V}(\phi_i,\phi_i;\phi_i,\phi_i)=U V ( ϕ i , ϕ i ; ϕ i , ϕ i ) = U C e − .18 D Ce^{-.18D} C e − .18 D 1 / 2 1/2 1/2 U n i ↑ n i ↓ Un_{i\uparrow}n_{i\downarrow} U n i ↑ n i ↓
∥ W e f f − U ∑ i n i ↑ n i ↓ ∥ ≤ C ( m 5 D + m 4 e − .18 D + m 2 D ) ≤ C m 5 D . \left\lVert W_{\mathrm{eff}}-U\sum_i n_{i\uparrow}n_{i\downarrow}\right\rVert\le C\left(\frac{m^5}{D}+m^4e^{-.18D}+\frac{m^2}{D}\right)\le\frac{Cm^5}{D}. W eff − U i ∑ n i ↑ n i ↓ ≤ C ( D m 5 + m 4 e − .18 D + D m 2 ) ≤ D C m 5 . The simulation estimate in physical energy units
Proposition 5.6 (Simulation of the Hubbard spectral minimum). Let E 0 = inf Spec H E_0=\inf\operatorname{Spec} H E 0 = inf Spec H E H u b ( t ) E_{\mathrm{Hub}}(t) E Hub ( t ) (7) . The construction satisfies
∣ E 0 Z ∗ + m Z ∗ 2 − E H u b ( t ) ∣ ≤ m δ 1 + C m 5 D + Z ∗ e − 1.2 D (119) \left|\frac{E_0}{Z_*}+\frac{mZ_*}{2}-E_{\mathrm{Hub}}(t)\right|\le m\delta_1+\frac{Cm^5}{D}+Z_*e^{-1.2D} \tag*{(119)} Z ∗ E 0 + 2 m Z ∗ − E Hub ( t ) ≤ m δ 1 + D C m 5 + Z ∗ e − 1.2 D (119) ≤ C [ m 2 ϵ h o p + m 5 D + m e − .01 D + m 2 e − .1 D + m 2 e − .6 D + e − .2 D ] . (120) \le C\left[m^2\epsilon_{\mathrm{hop}}+\frac{m^5}{D}+me^{-.01D}+m^2e^{-.1D}
+m^2e^{-.6D}+e^{-.2D}\right]. \tag*{(120)} ≤ C [ m 2 ϵ hop + D m 5 + m e − .01 D + m 2 e − .1 D + m 2 e − .6 D + e − .2 D ] . (120) In particular, for every prescribed fixed inverse-polynomial error budget ϵ ∗ \epsilon_* ϵ ∗ ϵ ∗ \epsilon_* ϵ ∗
Proof. The exact compression, in the abstract site space, is
J m ∗ ( Z ∗ H ) J m = d Γ ( Z ∗ A ∗ A ) + W e f f . \mathcal{J}_m^*(Z_* \mathcal{H})\mathcal{J}_m=d\Gamma(Z_* \mathcal{A}^*\mathcal{A})+W_{\mathrm{eff}}. J m ∗ ( Z ∗ H ) J m = d Γ ( Z ∗ A ∗ A ) + W eff . The Hubbard hopping term is d Γ ( T ⊗ I 2 ) d\Gamma(T\otimes I_2) d Γ ( T ⊗ I 2 ) m m m m m m (95) and Lemma (99) therefore give
∥ J m ∗ ( Z ∗ H ) J m − H H u b ( t ) ∥ ≤ m δ 1 + C m 5 D . \left\lVert\mathcal{J}_m^*(Z_*\mathcal{H})\mathcal{J}_m-H_{\mathrm{Hub}}(t)\right\rVert\le m\delta_1+\frac{Cm^5}{D}. ∥ J m ∗ ( Z ∗ H ) J m − H Hub ( t ) ∥ ≤ m δ 1 + D C m 5 . For finite self-adjoint matrices, the difference of their spectral minima is bounded by the operator norm difference, by the variational principle. Lemma (98) adds at most Z ∗ e − 1.2 D Z_*e^{-1.2D} Z ∗ e − 1.2 D Ran Π \operatorname{Ran}\Pi Ran Π
Z ∗ inf Spec H = E 0 Z ∗ + m Z ∗ 2 . Z_*\inf\operatorname{Spec}\mathcal{H}=\frac{E_0}{Z_*}+\frac{mZ_*}{2}. Z ∗ inf Spec H = Z ∗ E 0 + 2 m Z ∗ . This proves (119) . Substitute (101) and Z ∗ ≤ 2 e D Z_*\le2e^D Z ∗ ≤ 2 e D (120) .
The right side involves only fixed polynomial factors in m m m D D D (99) and Proposition 4.8 allow the hopping error to be made an arbitrarily small fixed inverse polynomial. The simultaneous choices, their order, and polynomial bit cost are specified in Section 6. No selected-orbital assumption has been imposed on the spectral infimum in this comparison.
Parameters, computation, and the final promise
We now put the constructions in their algorithmic order. This matters because the tail formulas are proved for tuned wells, whereas the positions must be computed before the tuning is performed. The surrogate fields in Section 4 remove this apparent dependence.
Let n ≥ 2 n \ge2 n ≥ 2 E i n = E ( H i n ) E_{\mathrm{in}} = E(H_{\mathrm{in}}) E in = E ( H in )
δ s = b s − a s , γ = min { 1 , δ s } , α = η 2 U . \delta_s = b_s - a_s,\qquad\gamma= \min\{1,\delta_s\},\qquad\alpha= \frac{\eta^2}{U}. δ s = b s − a s , γ = min { 1 , δ s } , α = U η 2 . Use the scales and rational hopping approximation of Theorem 2.6. They give
∣ E H u b ( t ) − α ( E i n − C s ) ∣ ≤ 3 α γ 64 , C s = 3 k Δ + 3 2 ∑ e v e 2 + ∑ { i , j } ∈ C K i j . (121) \left|E_{\mathrm{Hub}}(t) - \alpha(E_{\mathrm{in}} - C_s)\right| \le\frac{3\alpha\gamma}{64},\qquad C_s = 3k\Delta+ \frac{3}{2}\sum_e v_e^2 + \sum_{\{i,j\}\in\mathcal{C}} K_{ij}. \tag*{(121)} ∣ E Hub ( t ) − α ( E in − C s ) ∣ ≤ 64 3 α γ , C s = 3 k Δ + 2 3 e ∑ v e 2 + { i , j } ∈ C ∑ K ij . (121) All parameters chosen so far have polynomial magnitude or inverse-polynomial lower bounds as appropriate. Their binary descriptions have polynomial length. The scale α \alpha α C s C_s C s E i n E_{\mathrm{in}} E in
Set the requested continuum error to
ϵ ∗ = α γ 64 . \epsilon_* = \frac{\alpha\gamma}{64}. ϵ ∗ = 64 α γ . We verify that a single deterministic polynomial choice of D D D
S ≤ C n b ( 1 + log D ) S \le Cn^b(1+\log D) S ≤ C n b ( 1 + log D ) for a fixed b b b m m m ∣ log t i j ∣ |\log t_{ij}| ∣ log t ij ∣ n n n b b b m ≤ C n b m \le Cn^b m ≤ C n b
e ( D , τ ) = m log D + m 2 D + m S D + S 2 D + S D − 1 / 4 + τ + e − c D . (122) e(D,\tau) = \frac{m\log D + m^2}{D} + \frac{mS}{D} + \frac{S^2}{D} + SD^{-1/4} + \tau+ e^{-c\sqrt{D}}. \tag*{(122)} e ( D , τ ) = D m log D + m 2 + D m S + D S 2 + S D − 1/4 + τ + e − c D . (122) The other errors are poly ( m ) / D \operatorname{poly}(m)/D poly ( m ) / D h , r h,r h , r C n h Cn^h C n h ϵ ∗ ≥ c n − r \epsilon_* \ge cn^{-r} ϵ ∗ ≥ c n − r m 2 + m 5 ≤ C n h m^2 + m^5 \le Cn^h m 2 + m 5 ≤ C n h (120) are then bounded by C n h ( e ( D , τ ) + D − 1 ) Cn^h(e(D,\tau) + D^{-1}) C n h ( e ( D , τ ) + D − 1 )
Choose an integer K K K
K > 4 ( b + h + r + 2 ) , K > 4(b+h+r+2), K > 4 ( b + h + r + 2 ) , and put
D = A ( n + 2 ) K , τ = c ∗ ( n + 2 ) − h − r − 2 , D = A(n+2)^K,\qquad\tau= c_*(n+2)^{-h-r-2}, D = A ( n + 2 ) K , τ = c ∗ ( n + 2 ) − h − r − 2 , where A A A c ∗ > 0 c_* > 0 c ∗ > 0 (6.3) do not make this choice circular: K K K D D D A A A log A \log A log A n n n (6.5) . In particular,
n h S D − 1 / 4 = o ( n − r ) , n h S 2 / D = o ( n − r ) , m + S = o ( D 1 / 4 ) . n^{h}SD^{-1/4}=o(n^{-r}),\qquad n^{h}S^{2}/D=o(n^{-r}),\qquad m+S=o(D^{1/4}). n h S D − 1/4 = o ( n − r ) , n h S 2 / D = o ( n − r ) , m + S = o ( D 1/4 ) . Here the little-oh statements describe the uniform large-scale estimates used to choose the fixed constants; increasing A A A n n n (122) and the Coulomb-compression error by their assigned fractions of (6.2) . No search for a successful D D D
It is important to keep the energy units explicit. After dilation and subtraction of the isolated-well ground energy,
H = H / Z ∗ 2 + m / 2 , Z ∗ H = H / Z ∗ + m Z ∗ / 2. (123) \mathcal{H}=H/Z_{*}^{2}+m/2,\qquad Z_{*}\mathcal{H}=H/Z_{*}+mZ_{*}/2. \tag*{(123)} H = H / Z ∗ 2 + m /2 , Z ∗ H = H / Z ∗ + m Z ∗ /2. (123) The exponential margins that are needed after multiplication by Z ∗ ≍ e D Z_{*}\asymp e^{D} Z ∗ ≍ e D
Table 1 (PDF p. 44).
The polynomial multiplicities are retained in (120) . The positive exponential margins allow all these terms to fit their assigned error budgets by increasing the fixed A A A e − .75 D e^{-.75D} e − .75 D Z ∗ Z_{*} Z ∗
∣ E 0 / Z ∗ + m Z ∗ / 2 − α ( E i n − C s ) ∣ ≤ α γ 16 ≤ α δ s 16 . (124) \left|E_{0}/Z_{*}+mZ_{*}/2-\alpha(E_{\mathrm{in}}-C_{s})\right|\leq\frac{\alpha\gamma}{16}\leq\frac{\alpha\delta_{s}}{16}. \tag*{(124)} ∣ E 0 / Z ∗ + m Z ∗ /2 − α ( E in − C s ) ∣ ≤ 16 α γ ≤ 16 α δ s . (124) The deterministic construction and its bit cost
First check the source encoding and the fixed polynomial bounds in Theorem 2.1, including the inverse-gap bound. If a check fails, output the fixed instance with one electron, one unit-charge nucleus at the origin, and thresholds − 1 / 2 -1/2 − 1/2 1 / 2 1/2 1/2
Finite models. Read the signed rational Heisenberg instance. Form the ancillary graph, the algebraic reference positions p i p_i p i C \mathcal{C} C Δ , η \Delta,\eta Δ , η t i j t_{ij} t ij
Charge scale. Choose D D D (6.6) and compute a positive integer Z ∗ Z_{*} Z ∗
e D / 2 ≤ Z ∗ ≤ 2 e D . e^{D}/2\leq Z_{*}\leq2e^{D}. e D /2 ≤ Z ∗ ≤ 2 e D . This does not require an exact comparison with a transcendental number. For example, compute the rational sum s D = ∑ j = 0 4 D D j / j ! s_{D}=\sum_{j=0}^{4D}D^{j}/j! s D = ∑ j = 0 4 D D j / j ! Z ∗ = ⌊ s D ⌋ Z_{*}=\lfloor s_{D}\rfloor Z ∗ = ⌊ s D ⌋ 1 / 4 1/4 1/4 j ! ≥ ( j / e ) j j! \ge(j/e)^j j ! ≥ ( j / e ) j 0 < e D − s D < 1 0 < e^D - s_D < 1 0 < e D − s D < 1 D D D Z ∗ ( 1 − 500 m / D ) Z_*(1-500m/D) Z ∗ ( 1 − 500 m / D ) Z ∗ 1000 m / D Z_*1000m/D Z ∗ 1000 m / D
Positions. At the reference arrangement X i 0 = D p i X_i^0 = Dp_i X i 0 = D p i
d i j = log ( Z ∗ T i j ( X 0 ) / t i j ) d_{ij} = \log(Z_*T_{ij}(X^0)/t_{ij}) d ij = log ( Z ∗ T ij ( X 0 ) / t ij ) to the prescribed inverse-polynomial error. Apply the explicit right inverse from Theorem 2.5 and round the resulting scaled centers to rational X i X_i X i s i = X i − D p i s_i = X_i - Dp_i s i = X i − D p i W i ∗ W_i^* W i ∗
Well tuning. With these final rational centers and integer charges fixed, apply the certified one-well oracle and the contraction iteration of Theorem 3.7. This produces rational ρ i ∈ [ 1 , 4 ] \rho_i \in[1,4] ρ i ∈ [ 1 , 4 ] ∣ λ i + 1 / 2 ∣ ≤ e − 1.01 D |\lambda_i + 1/2| \le e^{-1.01D} ∣ λ i + 1/2∣ ≤ e − 1.01 D
Physical instance. Output the 2 m 2m 2 m
R i , 0 = X i / Z ∗ , R i , 1 = ( X i + ρ i e z ) / Z ∗ R_{i,0} = X_i/Z_*, \qquad R_{i,1} = (X_i + \rho_i e_z)/Z_* R i , 0 = X i / Z ∗ , R i , 1 = ( X i + ρ i e z ) / Z ∗ with their positive integer charges, and output N = m N=m N = m
Each step is classical and deterministic. The one-well Ritz spaces have dimension polynomial in D D D
The output has polynomial length. Each nuclear charge has O ( D + log n ) O(D+\log n) O ( D + log n ) O ( D ) O(D) O ( D ) Z ∗ Z_* Z ∗ D − O ( S ) D-O(S) D − O ( S ) D D D m m m n n n N N N
Rational thresholds and preservation of both promises
Compute a rational C ~ s \widetilde{C}_s C s
∣ C ~ s − C s ∣ ≤ δ s / 64. (125) |\widetilde{C}_s-C_s| \le\delta_s/64. \tag*{(125)} ∣ C s − C s ∣ ≤ δ s /64. (125) Polynomially many square-root approximations suffice, since C s C_s C s δ s \delta_s δ s
F ~ ( z ) = − m Z ∗ 2 / 2 + Z ∗ α ( z − C ~ s ) . (126) \widetilde{F}(z) = -mZ_*^2/2 + Z_*\alpha(z-\widetilde{C}_s). \tag*{(126)} F ( z ) = − m Z ∗ 2 /2 + Z ∗ α ( z − C s ) . (126) and set
a = F ~ ( a s + δ s / 4 ) , b = F ~ ( b s − δ s / 4 ) . (127) a = \widetilde{F}(a_s + \delta_s/4), \qquad b = \widetilde{F}(b_s - \delta_s/4). \tag*{(127)} a = F ( a s + δ s /4 ) , b = F ( b s − δ s /4 ) . (127) All the quantities in these formulas are rational and computable with polynomial bit cost. The possibly enormous common offset − m Z ∗ 2 / 2 -mZ_*^2/2 − m Z ∗ 2 /2 (124) and (125) ,
∣ E 0 − F ~ ( E i n ) ∣ ≤ 5 64 Z ∗ α δ s < 1 8 Z ∗ α δ s . (128) \lvert E_0 - \widetilde{F}(E_{\mathrm{in}}) \rvert\le\frac{5}{64}Z_*\alpha\delta_s < \frac{1}{8}Z_*\alpha\delta_s. \tag*{(128)} ∣ E 0 − F ( E in )∣ ≤ 64 5 Z ∗ α δ s < 8 1 Z ∗ α δ s . (128) Since F ~ \widetilde{F} F
E 0 ≤ F ~ ( a s ) + 1 8 Z ∗ α δ s ≤ a − 1 8 Z ∗ α δ s < a , E_0 \le\widetilde{F}(a_s) + \frac{1}{8}Z_*\alpha\delta_s \le a - \frac{1}{8}Z_*\alpha\delta_s < a, E 0 ≤ F ( a s ) + 8 1 Z ∗ α δ s ≤ a − 8 1 Z ∗ α δ s < a , and a NO source instance satisfies
E 0 ≥ F ~ ( b s ) − 1 8 Z ∗ α δ s ≥ b + 1 8 Z ∗ α δ s > b . E_0 \ge\widetilde{F}(b_s) - \frac{1}{8}Z_*\alpha\delta_s \ge b + \frac{1}{8}Z_*\alpha\delta_s > b. E 0 ≥ F ( b s ) − 8 1 Z ∗ α δ s ≥ b + 8 1 Z ∗ α δ s > b . These inequalities concern the exact continuum spectral infimum by Theorem 5.6, rather than only a projected eigenvalue.
The threshold separation is exactly
b − a = 1 2 Z ∗ α δ s . (129) b - a = \frac{1}{2}Z_*\alpha\delta_s. \tag*{(129)} b − a = 2 1 Z ∗ α δ s . (129) The reciprocal of α δ s \alpha\delta_s α δ s n n n Z ∗ ≥ e D / 2 Z_* \ge e^{D/2} Z ∗ ≥ e D /2 (129) at least one for every input. Since the output length L ≥ 1 L \ge1 L ≥ 1 b − a ≥ L − 1 b-a \ge L^{-1} b − a ≥ L − 1 L L L
We have constructed a legal nuclear instance in deterministic polynomial time, preserving YES and NO and permitting all spin sectors and all continuum states. This proves Theorem 1.1.
