Introduction
The spectrum of the planar Neumann–Poincaré (NP) operator is unchanged by extended Möbius transformations. Do these transformations account for all domains with the same spectrum? We show that they do not, even for analytic, strictly convex domains arbitrarily close to the disk. We also show that finitely many eigenvalues can determine a domain stably within a prescribed finite-dimensional family.
Let D ⊂ R 2 D \subset\mathbb{R}^{2} D ⊂ R 2 ν \nu ν
K D ∗ [ φ ] ( x ) = 1 2 π ∫ ∂ D ( x − y ) ⋅ ν x ∣ x − y ∣ 2 φ ( y ) d s y , x ∈ ∂ D . (1) K_D^*[\varphi](x)=\frac{1}{2\pi}\int_{\partial D}\frac{(x-y)\cdot\nu_x}{|x-y|^2}\varphi(y)\,\mathrm{d}s_y,\qquad x\in\partial D. \tag*{(1)} K D ∗ [ φ ] ( x ) = 2 π 1 ∫ ∂ D ∣ x − y ∣ 2 ( x − y ) ⋅ ν x φ ( y ) d s y , x ∈ ∂ D . (1) The single-layer potential has kernel ( 2 π ) − 1 log ∣ x − y ∣ (2\pi)^{-1}\log|x-y| ( 2 π ) − 1 log ∣ x − y ∣ ( 1 2 I + K D ∗ ) φ (\frac{1}{2}I+K_D^*)\varphi ( 2 1 I + K D ∗ ) φ ( − 1 2 I + K D ∗ ) φ (-\frac{1}{2}I+K_D^*)\varphi ( − 2 1 I + K D ∗ ) φ H 0 − 1 / 2 ( ∂ D ) H^{-1/2}_0(\partial D) H 0 − 1/2 ( ∂ D ) 1 / 2 1/2 1/2 μ \mu μ − μ -\mu − μ [15 ] . We denote the positive eigenvalues in nonincreasing order by λ j ( D ) \lambda_j(D) λ j ( D )
Schiffer connected this spectrum with conformal mapping through his work on Fredholm eigenvalues [18 , 19 ] . The relation between the classical Fredholm parameter and an NP eigenvalue depends on the normalisation of the integral equation. NP eigenvalues also describe quasistatic resonances of dielectric inclusions; see [3 , 2 ] . Recovery and design from partial spectral data were studied in [1 ] , and reconstruction from plasmonic resonances for particular families in [4 ] . Shape derivatives and real-analytic dependence of symmetric functions of isolated nonzero NP eigenvalue clusters are established in [9 ] .
The counterexample is a pair of degree-seven polynomial lemniscates. A factorisation of the difference of their defining polynomials gives a correspondence between the boundaries. Its invertible incidence matrix, combined with explicit smooth multipliers, intertwines the NP operators. An algebraic argument then rules out holomorphic and antiholomorphic Möbius equivalence. The boundaries are analytic, strictly convex and real algebraic, and approach the circle after rescaling.
For the positive results we fix capacity and translation by taking exterior conformal maps with leading coefficient one and constant coefficient zero. The first m m m m m m m m m
The proof begins with the Grunsky operator. Its linear part is a finite weighted Hankel matrix, and we find directions in which the characteristic coefficients have an invertible differential and the singular values are distinct and nonzero. The nonlinear remainder is quadratic in Hilbert–Schmidt norm, with a derivative of first order. These estimates control the derivatives of the leading eigenvalues even though the spectral gaps shrink linearly at the disk. On a narrow convex cone the spectral differential stays close to one fixed invertible matrix. Integration along line segments then gives injectivity and the inverse Lipschitz estimate throughout the cone.
The resulting estimates compare parameter distance, weighted Laurent coefficient norms, Hausdorff distance, and distances between complete ordered spectra. Forward estimates hold on a small parameter ball; inverse estimates require the nondegenerate cone. For two coefficients, an explicit inverse of the leading spectral map gives a full sector. For three, exact root bounds give a cone about ( 1 , 1 , 1 ) (1,1,1) ( 1 , 1 , 1 )
Why, then, can polarization measurements determine a domain? They contain more than eigenvalues. For a harmonic polynomial H H H G G G
M H , G ( λ ; D ) = ∫ ∂ D G ( λ I − K D ∗ ) − 1 [ ∂ ν H ] d s , (2) M_{H,G}(\lambda;D)=\int_{\partial D}G(\lambda I-K_D^*)^{-1}[\partial_\nu H]\,\mathrm{d}s, \tag*{(2)} M H , G ( λ ; D ) = ∫ ∂ D G ( λ I − K D ∗ ) − 1 [ ∂ ν H ] d s , (2) where the inverse acts on H 0 − 1 / 2 ( ∂ D ) H_0^{-1/2}(\partial D) H 0 − 1/2 ( ∂ D ) λ \lambda λ k ≠ 1 k\ne1 k = 1 λ = ( k + 1 ) / ( 2 ( k − 1 ) ) \lambda=(k+1)/(2(k-1)) λ = ( k + 1 ) / ( 2 ( k − 1 )) (2) contains eigenvalues together with geometric excitation–observation weights. For positive conductivities, Ammari and Kang [3 Theorem 4.9] recover the inclusion and contrast from all generalised polarization tensors (GPTs). Their proof recovers a Dirichlet-to-Neumann map and uses inverse conductivity uniqueness [11 ] . Uniqueness from harmonic combinations alone is stated in [5 Theorem 3.1] ; identification and reconstruction of algebraic boundaries from finitely many GPTs are studied in [5 , 6 ] . We give two direct proofs for complete data: weighted spectral data determine all mixed area moments, and full uncontracted GPTs at one fixed contrast determine an analytic Jordan domain and its contrast. The second proof uses all polynomial observations, a richer data set than harmonic observations alone. Spectral uniqueness and polarization uniqueness therefore ask different questions.
Section 2 constructs the counterexample. Section 3 proves finite-dimensional recovery and stability, and § 6 works out the two- and three-parameter cases. § 7 discusses questions left open. Appendix § 8 studies the two-moment map and its critical curve; Appendix § 9 gives the polarization uniqueness proofs; Appendix § 10 develops further uniqueness classes.
An isospectral pair of polynomial lemniscates
A degree-seven polynomial correspondence supplies the counterexample. The proof has three parts: establish the geometry of the lemniscates, construct a similarity of their NP operators, and rule out extended Möbius equivalence. Polynomial difference factorisations and correspondences are treated in [8 , 10 ] ; transplantation for planar Dirichlet spectra is developed in [7 ] .
An extended Möbius transformation is a Möbius map, possibly followed by complex conjugation. We require its pole to lie off the boundary and take the transformed domain to be the bounded component enclosed by the image boundary.
Let
d = i 7 , α = − 1 + d 2 , β = − 5 + d 14 , d = i\sqrt{7}, \qquad\alpha= -\frac{1+d}{2}, \qquad\beta= \frac{-5+d}{14}, d = i 7 , α = − 2 1 + d , β = 14 − 5 + d , and form
P ( z ) = z 7 + α z 4 + β z + i , Q ( z ) = z 7 + α ˉ z 4 + β ˉ z + i . (3) P(z) = z^7 + \alpha z^4 + \beta z + i,\qquad Q(z) = z^7 + \bar{\alpha}z^4 + \bar{\beta}z + i. \tag*{(3)} P ( z ) = z 7 + α z 4 + β z + i , Q ( z ) = z 7 + α ˉ z 4 + β ˉ z + i . (3) For R ≥ 4096 R \ge4096 R ≥ 4096 D P = { ∣ P ∣ < R } D_P = \{|P| < R\} D P = { ∣ P ∣ < R } D Q = { ∣ Q ∣ < R } D_Q = \{|Q| < R\} D Q = { ∣ Q ∣ < R }
Theorem 2.1 (Isospectral polynomial lemniscates). The domains D P D_P D P D Q D_Q D Q L 2 L^2 L 2 H s ( ∂ D P ) H^s(\partial D_P) H s ( ∂ D P ) H s ( ∂ D Q ) H^s(\partial D_Q) H s ( ∂ D Q ) s ∈ R s \in\mathbb{R} s ∈ R z ↦ R − 1 / 7 z z \mapsto R^{-1/7}z z ↦ R − 1/7 z C k C^k C k R → ∞ R \to\infty R → ∞
Figure 1 shows the pair at the threshold.
Figure 1. The counterexample at R = 4096 R = 4096 R = 4096 R − 1 / 7 R^{-1/7} R − 1/7 10 3 ( r P ( θ ) − 1 ) 10^3(r_P(\theta)-1) 1 0 3 ( r P ( θ ) − 1 ) 10 3 ( r Q ( θ ) − 1 ) 10^3(r_Q(\theta)-1) 1 0 3 ( r Q ( θ ) − 1 )
Proof. 1. Boundary geometry. Let p ( z ) = z 7 + A z 4 + B z + i p(z) = z^7 + Az^4 + Bz + i p ( z ) = z 7 + A z 4 + B z + i (3) . Since ∣ A ∣ < 3 / 2 |A| < 3/2 ∣ A ∣ < 3/2 ∣ B ∣ < 1 / 2 |B| < 1/2 ∣ B ∣ < 1/2 ∣ z ∣ ≤ 3 |z| \le3 ∣ z ∣ ≤ 3
∣ p ( z ) ∣ ≤ 3 7 + 3 2 3 4 + 1 2 3 + 1 = 2311 < 4096. |p(z)| \le3^7 + \frac{3}{2}3^4 + \frac{1}{2}3 + 1 = 2311 < 4096. ∣ p ( z ) ∣ ≤ 3 7 + 2 3 3 4 + 2 1 3 + 1 = 2311 < 4096. For ∣ z ∣ ≥ 3 |z| \ge3 ∣ z ∣ ≥ 3
p = z 7 ( 1 + e ) , p ′ = 7 z 6 ( 1 + f ) , p ′ ′ = 42 z 5 ( 1 + g ) . p = z^{7}(1+e), \qquad p' = 7z^{6}(1+f), \qquad p'' = 42z^{5}(1+g). p = z 7 ( 1 + e ) , p ′ = 7 z 6 ( 1 + f ) , p ′′ = 42 z 5 ( 1 + g ) . with
e = A z − 3 + B z − 6 + i z − 7 , f = 4 7 A z − 3 + 1 7 B z − 6 , g = 2 7 A z − 3 . e = Az^{-3}+Bz^{-6}+iz^{-7}, \qquad f = \frac{4}{7}Az^{-3}+\frac{1}{7}Bz^{-6}, \qquad g = \frac{2}{7}Az^{-3}. e = A z − 3 + B z − 6 + i z − 7 , f = 7 4 A z − 3 + 7 1 B z − 6 , g = 7 2 A z − 3 . Then
∣ e ∣ < 3 50 , ∣ f ∣ < 1 30 , ∣ g ∣ < 1 60 . |e| < \frac{3}{50}, \qquad|f| < \frac{1}{30}, \qquad|g| < \frac{1}{60}. ∣ e ∣ < 50 3 , ∣ f ∣ < 30 1 , ∣ g ∣ < 60 1 . Thus p , p ′ p,p' p , p ′ R e ( z p ′ / p ) > 0 \mathop{\mathrm{Re}}(zp'/p)>0 Re ( z p ′ / p ) > 0 ∣ p ∣ |p| ∣ p ∣ R R R
To check convexity, parametrise the curve locally by p ( z ( t ) ) = R e i t p(z(t))=Re^{it} p ( z ( t )) = R e i t
d d t arg z ′ ( t ) = R e ( 1 − p p ′ ′ ( p ′ ) 2 ) . \frac{\mathrm{d}}{\mathrm{d}t}\arg z'(t)=\mathop{\mathrm{Re}}\left(1-\frac{pp''}{(p')^2}\right). d t d arg z ′ ( t ) = Re ( 1 − ( p ′ ) 2 p p ′′ ) . The estimates above give
∣ ( 1 + e ) ( 1 + g ) ( 1 + f ) 2 − 1 ∣ < 1309 8410 < 1 6 , \left|\frac{(1+e)(1+g)}{(1+f)^2}-1\right|<\frac{1309}{8410}<\frac{1}{6}, ( 1 + f ) 2 ( 1 + e ) ( 1 + g ) − 1 < 8410 1309 < 6 1 , so
d d t arg z ′ ( t ) > 1 7 − 6 7 1309 8410 = 278 29435 > 0. \frac{\mathrm{d}}{\mathrm{d}t}\arg z'(t)>\frac{1}{7}-\frac{6}{7}\frac{1309}{8410}=\frac{278}{29435}>0. d t d arg z ′ ( t ) > 7 1 − 7 6 8410 1309 = 29435 278 > 0. The curvature is positive. The parametrisation is local because one circuit of the boundary maps to seven circuits of the target circle. Every target value τ \tau τ ∣ τ ∣ ≥ 4096 |\tau|\ge4096 ∣ τ ∣ ≥ 4096
Rescale by z = R 1 / 7 u z=R^{1/7}u z = R 1/7 u P P P
∣ u 7 + α R − 3 / 7 u 4 + β R − 6 / 7 u + i R − 1 ∣ = 1 , \left|u^7+\alpha R^{-3/7}u^4+\beta R^{-6/7}u+iR^{-1}\right|=1, u 7 + α R − 3/7 u 4 + β R − 6/7 u + i R − 1 = 1 , and the equation for Q Q Q α \alpha α β \beta β ∣ u 7 ∣ = 1 |u^7|=1 ∣ u 7 ∣ = 1 ∣ u ∣ = 1 |u|=1 ∣ u ∣ = 1 C k C^k C k
Similarity of the NP operators
The construction rests on the exact factorisation
P ( x ) − Q ( y ) = H ( x , y ) G ( x , y ) , (4) P(x)-Q(y)=H(x,y)G(x,y), \tag*{(4)} P ( x ) − Q ( y ) = H ( x , y ) G ( x , y ) , (4) where
H ( x , y ) = x 3 + 1 + d 2 x 2 y + − 1 + d 2 x y 2 − y 3 − d 7 , H(x,y)=x^3+\frac{1+d}{2}x^2y+\frac{-1+d}{2}xy^2-y^3-\frac{d}{7}, H ( x , y ) = x 3 + 2 1 + d x 2 y + 2 − 1 + d x y 2 − y 3 − 7 d , G ( x , y ) = x 4 − 1 + d 2 x 3 y − x 2 y 2 + − 1 + d 2 x y 3 + y 4 − 7 + 5 d 14 x + − 7 + 5 d 14 y . G(x,y)=x^4-\frac{1+d}{2}x^3y-x^2y^2+\frac{-1+d}{2}xy^3+y^4-\frac{7+5d}{14}x+\frac{-7+5d}{14}y. G ( x , y ) = x 4 − 2 1 + d x 3 y − x 2 y 2 + 2 − 1 + d x y 3 + y 4 − 14 7 + 5 d x + 14 − 7 + 5 d y . Over a common regular value t t t x x x P ( x ) = t P(x)=t P ( x ) = t y y y Q ( y ) = t Q(y)=t Q ( y ) = t H ( x , y ) = 0 H(x,y)=0 H ( x , y ) = 0 7 × 7 7\times7 7 × 7 N N N y y y H H H P ′ = H x G P'=H_xG P ′ = H x G − Q ′ = H y G -Q'=H_yG − Q ′ = H y G G = 0 G=0 G = 0 H x = 0 H_x=0 H x = 0 H y = 0 H_y=0 H y = 0
Near infinity, label the roots so that x j ∼ t 1 / 7 ζ j x_j\sim t^{1/7}\zeta^j x j ∼ t 1/7 ζ j y k ∼ t 1 / 7 ζ k y_k\sim t^{1/7}\zeta^k y k ∼ t 1/7 ζ k ζ = e 2 π i / 7 \zeta=e^{2\pi i/7} ζ = e 2 π i /7
ζ + ζ 2 + ζ 4 = − 1 + d 2 \zeta+\zeta^2+\zeta^4=\frac{-1+d}{2} ζ + ζ 2 + ζ 4 = 2 − 1 + d gives
H 3 ( X , 1 ) = ∏ q ∈ { 3 , 5 , 6 } ( X − ζ q ) . H_3(X,1)=\prod_{q\in\{3,5,6\}}(X-\zeta^q). H 3 ( X , 1 ) = q ∈ { 3 , 5 , 6 } ∏ ( X − ζ q ) . where H 3 H_3 H 3 H H H j − k ∈ { 3 , 5 , 6 } j-k\in\{3,5,6\} j − k ∈ { 3 , 5 , 6 }
N N T = N T N = 2 I + J 7 , N − 1 = 1 2 N T − 1 6 J 7 . (5) NN^{\mathsf T}=N^{\mathsf T}N=2I+J_7,\qquad N^{-1}=\frac{1}{2}N^{\mathsf T}-\frac{1}{6}J_7. \tag*{(5)} N N T = N T N = 2 I + J 7 , N − 1 = 2 1 N T − 6 1 J 7 . (5) Now continue the roots along a path avoiding the finitely many critical values of P P P Q Q Q H H H P ′ = H x G P'=H_xG P ′ = H x G − Q ′ = H y G -Q'=H_yG − Q ′ = H y G H x H y G ≠ 0 H_xH_yG\ne0 H x H y G = 0 N N N (5) holds over every regular value, in particular on ∣ t ∣ = R |t|=R ∣ t ∣ = R J 7 J_7 J 7
To pass from incidences to integral operators, we use
∑ H ( x , y ) = 0 1 P ′ ( x ) ( x − x ′ ) = 1 Q ′ ( y ) ∑ H ( x ′ , y ′ ) = 0 1 y − y ′ . (6) \sum_{H(x,y)=0}\frac{1}{P'(x)(x-x')}=\frac{1}{Q'(y)}\sum_{H(x',y')=0}\frac{1}{y-y'}. \tag*{(6)} H ( x , y ) = 0 ∑ P ′ ( x ) ( x − x ′ ) 1 = Q ′ ( y ) 1 H ( x ′ , y ′ ) = 0 ∑ y − y ′ 1 . (6) Locally, H ( X , y ) = ∏ j = 1 3 ( X − x j ( y ) ) H(X,y)=\prod_{j=1}^{3}(X-x_j(y)) H ( X , y ) = ∏ j = 1 3 ( X − x j ( y )) P ( x j ( y ) ) = Q ( y ) P(x_j(y))=Q(y) P ( x j ( y )) = Q ( y ) x j ′ = Q ′ / P ′ ( x j ) x_j'=Q'/P'(x_j) x j ′ = Q ′ / P ′ ( x j ) ∂ y log H ( x ′ , y ) \partial_y\log H(x',y) ∂ y log H ( x ′ , y ) Q ′ ( y ) Q'(y) Q ′ ( y ) y y y − 1 -1 − 1
Let M P M_P M P M Q M_Q M Q ∣ P ′ ∣ |P'| ∣ P ′ ∣ ∣ Q ′ ∣ |Q'| ∣ Q ′ ∣
( V φ ) ( y ) = ∣ Q ′ ( y ) ∣ ∑ H ( x , y ) = 0 φ ( x ) ∣ P ′ ( x ) ∣ (7) (V\varphi)(y)=|Q'(y)|\sum_{H(x,y)=0}\frac{\varphi(x)}{|P'(x)|} \tag*{(7)} ( V φ ) ( y ) = ∣ Q ′ ( y ) ∣ H ( x , y ) = 0 ∑ ∣ P ′ ( x ) ∣ φ ( x ) (7) is M Q N M P − 1 M_QNM_P^{-1} M Q N M P − 1 (5) :
( V − 1 ψ ) ( x ) = ∣ P ′ ( x ) ∣ [ 1 2 ∑ H ( x , y ) = 0 ψ ( y ) ∣ Q ′ ( y ) ∣ − 1 6 ∑ Q ( y ) = P ( x ) ψ ( y ) ∣ Q ′ ( y ) ∣ ] . (V^{-1}\psi)(x)=|P'(x)|\left[\frac{1}{2}\sum_{H(x,y)=0}\frac{\psi(y)}{|Q'(y)|}-\frac{1}{6}\sum_{Q(y)=P(x)}\frac{\psi(y)}{|Q'(y)|}\right]. ( V − 1 ψ ) ( x ) = ∣ P ′ ( x ) ∣ 2 1 H ( x , y ) = 0 ∑ ∣ Q ′ ( y ) ∣ ψ ( y ) − 6 1 Q ( y ) = P ( x ) ∑ ∣ Q ′ ( y ) ∣ ψ ( y ) . The coverings are smooth and unbranched. Thus both maps, finite sums of smooth compositions with nonvanishing multipliers, are bounded on every H s H^s H s L 2 L^2 L 2
On the first boundary,
ν P ( x ) ∣ P ′ ( x ) ∣ = P ( x ) R P ′ ( x ) . \frac{\nu_P(x)}{|P'(x)|}=\frac{P(x)}{RP'(x)}. ∣ P ′ ( x ) ∣ ν P ( x ) = R P ′ ( x ) P ( x ) . Changing variables to t = P ( x ) t=P(x) t = P ( x ) s = P ( x ′ ) s=P(x') s = P ( x ′ )
1 2 π R ℜ t P ′ ( x ) ( x − x ′ ) \frac{1}{2\pi R}\Re\frac{t}{P'(x)(x-x')} 2 π R 1 ℜ P ′ ( x ) ( x − x ′ ) t together with the sum over P ( x ′ ) = s P(x')=s P ( x ′ ) = s ∣ d s ∣ |ds| ∣ d s ∣ (6) by t t t
K D Q ∗ V = V K D P ∗ . (8) K_{D_Q}^*V=VK_{D_P}^*. \tag*{(8)} K D Q ∗ V = V K D P ∗ . (8) Initially the target values s , t s,t s , t N N N ∫ ∂ D Q V φ d s = 3 ∫ ∂ D P φ d s \int_{\partial D_Q}V\varphi\,ds=3\int_{\partial D_P}\varphi\,ds ∫ ∂ D Q V φ d s = 3 ∫ ∂ D P φ d s
Möbius non-equivalence
For a polynomial A A A A # ( z ) = A ( z ‾ ) A^{\#}(z)=A(\overline{z}) A # ( z ) = A ( z ) A , B A,B A , B n n n A A A A A A n n n M M M { ∣ A ∣ = R } \{|A|=R\} { ∣ A ∣ = R } { ∣ B ∣ = R } \{|B|=R\} { ∣ B ∣ = R } B ∘ M = τ A B\circ M=\tau A B ∘ M = τ A ∣ τ ∣ = 1 |\tau|=1 ∣ τ ∣ = 1
If A A A R > 0 R>0 R > 0
A ( z ) A # ( w ) − R 2 A(z)A^{\#}(w)-R^{2} A ( z ) A # ( w ) − R 2 is irreducible in C [ z , w ] \mathbb{C}[z,w] C [ z , w ] [17 ] . Here Eisenstein’s criterion gives a short proof. Reverse the polynomial in z z z n = deg A n=\deg A n = deg A
u n [ A ( 1 / u ) A # ( w ) − R 2 ] . u^{n}\left[A(1/u)A^{\#}(w)-R^{2}\right]. u n [ A ( 1/ u ) A # ( w ) − R 2 ] . If A ( z ) = ∑ j = 0 n a j z j A(z)=\sum_{j=0}^{n}a_jz^j A ( z ) = ∑ j = 0 n a j z j w 0 w_0 w 0 A # A^{\#} A # u n u^n u n a 0 A # ( w ) − R 2 a_0A^{\#}(w)-R^2 a 0 A # ( w ) − R 2 w − w 0 w-w_0 w − w 0 w − w 0 w-w_0 w − w 0 a n A # ( w ) a_nA^{\#}(w) a n A # ( w ) ( w − w 0 ) 2 (w-w_0)^2 ( w − w 0 ) 2 C ( w ) [ u ] \mathbb{C}(w)[u] C ( w ) [ u ] A # A^{\#} A # a 0 A # − R 2 a_0A^{\#}-R^2 a 0 A # − R 2 u u u C [ z , w ] \mathbb{C}[z,w] C [ z , w ]
Let M ( z ) = ( α 0 z + β 0 ) / ( γ 0 z + η 0 ) M(z)=(\alpha_0z+\beta_0)/(\gamma_0z+\eta_0) M ( z ) = ( α 0 z + β 0 ) / ( γ 0 z + η 0 )
D ( z ) = ( γ 0 z + η 0 ) n , C ( z ) = D ( z ) B ( M ( z ) ) . D(z)=(\gamma_0z+\eta_0)^n,\qquad C(z)=D(z)B(M(z)). D ( z ) = ( γ 0 z + η 0 ) n , C ( z ) = D ( z ) B ( M ( z )) . The polynomial C ( z ) C # ( w ) − R 2 D ( z ) D # ( w ) C(z)C^{\#}(w)-R^2D(z)D^{\#}(w) C ( z ) C # ( w ) − R 2 D ( z ) D # ( w ) C , D C,D C , D n n n n n n C ( z ) C # ( w ) − R 2 D ( z ) D # ( w ) C(z)C^{\#}(w)-R^2D(z)D^{\#}(w) C ( z ) C # ( w ) − R 2 D ( z ) D # ( w ) span { C , D } \operatorname{span}\{C,D\} span { C , D } A ( z ) A # ( w ) − R 2 A(z)A^{\#}(w)-R^2 A ( z ) A # ( w ) − R 2 span { A , 1 } \operatorname{span}\{A,1\} span { A , 1 }
span { C , D } = span { A , 1 } . \operatorname{span}\{C,D\}=\operatorname{span}\{A,1\}. span { C , D } = span { A , 1 } . If γ 0 ≠ 0 \gamma_0\ne0 γ 0 = 0 A A A n n n M M M B ( M ( z ) ) = c 1 A ( z ) + c 0 B(M(z))=c_1A(z)+c_0 B ( M ( z )) = c 1 A ( z ) + c 0 c 1 ≠ 0 c_1\ne0 c 1 = 0 τ \tau τ ∣ τ ∣ = R |\tau|=R ∣ τ ∣ = R A ( z ) = τ A(z)=\tau A ( z ) = τ ∣ c 1 τ + c 0 ∣ = R |c_1\tau+c_0|=R ∣ c 1 τ + c 0 ∣ = R c 0 = 0 c_0=0 c 0 = 0 ∣ c 1 ∣ = 1 |c_1|=1 ∣ c 1 ∣ = 1
B ( α 0 z + β 0 ) = τ A ( z ) , ∣ τ ∣ = 1. (9) B(\alpha_0z+\beta_0)=\tau A(z),\qquad|\tau|=1. \tag*{(9)} B ( α 0 z + β 0 ) = τ A ( z ) , ∣ τ ∣ = 1. (9) We check these hypotheses for the polynomials in (3) . The equations P ′ ( z ) = 0 P'(z)=0 P ′ ( z ) = 0 Q ′ ( z ) = 0 Q'(z)=0 Q ′ ( z ) = 0 z 3 z^3 z 3 − 14 + 6 d -14+6d − 14 + 6 d − 14 − 6 d -14-6d − 14 − 6 d P P P P # P^{\#} P # P P P P # P^{\#} P #
A holomorphic equivalence gives Q ( α 0 z + β 0 ) = τ P ( z ) Q(\alpha_0z+\beta_0)=\tau P(z) Q ( α 0 z + β 0 ) = τ P ( z ) β 0 = 0 \beta_0=0 β 0 = 0 τ = 1 \tau=1 τ = 1
α 0 7 = 1 , α 0 ‾ α 0 4 = α , β ‾ α 0 = β , hence β α 2 = β ‾ α ‾ 2 . \alpha_0^7=1,\qquad\overline{\alpha_0}\alpha_0^4=\alpha,\qquad\overline{\beta}\alpha_0=\beta,\qquad\text{hence}\quad\frac{\beta}{\alpha^2}=\frac{\overline{\beta}}{\overline{\alpha}^{\,2}}. α 0 7 = 1 , α 0 α 0 4 = α , β α 0 = β , hence α 2 β = α 2 β . The two ratios, however, are ( 11 + d ) / 56 (11+d)/56 ( 11 + d ) /56 ( 11 − d ) / 56 (11-d)/56 ( 11 − d ) /56 (9) to P # ( z ) = z 7 + α ‾ z 4 + β ‾ z − i P^{\#}(z)=z^7+\overline{\alpha}z^4+\overline{\beta}z-i P # ( z ) = z 7 + α z 4 + β z − i τ = − 1 \tau=-1 τ = − 1 α 0 = − 1 \alpha_0=-1 α 0 = − 1 α ‾ = − α ‾ \overline{\alpha}=-\overline{\alpha} α = − α □ \square □
The counterexample leaves room for uniqueness within prescribed families. We now obtain it by making the higher conformal coefficients depend on finitely many free parameters.
Finite-dimensional uniqueness and stability
The family and the inverse theorem
To obtain stable recovery despite nonuniqueness in the unrestricted problem, we prescribe a finite-dimensional family. We use exterior conformal maps with coefficient of w w w
Fix m ∈ N m \in\mathbb{N} m ∈ N
F ( w ) = ∑ n ≥ m + 1 f n w − n , f n ∈ R , ∣ f n ∣ ≤ C F r ∗ n ( 0 < r ∗ < 1 ) . (10) F(w)=\sum_{n\geq m+1}f_n w^{-n},\qquad f_n\in\mathbb{R},\qquad|f_n|\leq C_F r_*^n\quad(0<r_*<1). \tag*{(10)} F ( w ) = n ≥ m + 1 ∑ f n w − n , f n ∈ R , ∣ f n ∣ ≤ C F r ∗ n ( 0 < r ∗ < 1 ) . (10) For real a = ( a 1 , … , a m ) \mathbf{a}=(a_1,\ldots,a_m) a = ( a 1 , … , a m )
Ψ a ( w ) = w + ∑ n = 1 m a n w − n + ∣ a ∣ 2 F ( w ) , ∣ a ∣ 2 = ∑ n = 1 m a n 2 . (11) \Psi_{\mathbf{a}}(w)=w+\sum_{n=1}^{m}a_n w^{-n}+|\mathbf{a}|^2F(w),\qquad|\mathbf{a}|^2=\sum_{n=1}^{m}a_n^2. \tag*{(11)} Ψ a ( w ) = w + n = 1 ∑ m a n w − n + ∣ a ∣ 2 F ( w ) , ∣ a ∣ 2 = n = 1 ∑ m a n 2 . (11) Let D a D_{\mathbf{a}} D a Ψ a ( { ∣ w ∣ = 1 } ) \Psi_{\mathbf{a}}(\{|w|=1\}) Ψ a ({ ∣ w ∣ = 1 }) K a = D a ‾ K_{\mathbf{a}}=\overline{D_{\mathbf{a}}} K a = D a Ψ a ( w ‾ ) = Ψ a ( w ) ‾ \Psi_{\mathbf{a}}(\overline{w})=\overline{\Psi_{\mathbf{a}}(w)} Ψ a ( w ) = Ψ a ( w ) D a D_{\mathbf{a}} D a
For small parameters, these maps define domains, as proved below. The first m m m (11) prescribes all higher coefficients. The choice F = 0 F=0 F = 0
For a unit vector v ∈ R m v\in\mathbb{R}^m v ∈ R m ε , δ > 0 \varepsilon,\delta>0 ε , δ > 0
C ( v , ε , δ ) = { t v + u : 0 < t < δ , u ⊥ v , ∣ u ∣ < ε t } . (12) C(v,\varepsilon,\delta)=\{tv+u:0<t<\delta,\ u\perp v,\ |u|<\varepsilon t\}. \tag*{(12)} C ( v , ε , δ ) = { t v + u : 0 < t < δ , u ⊥ v , ∣ u ∣ < εt } . (12) This truncated cone is convex and m m m m = 1 m=1 m = 1 t ≤ ∣ a ∣ ≤ 1 + ε 2 t t\leq|\mathbf{a}|\leq\sqrt{1+\varepsilon^2}\,t t ≤ ∣ a ∣ ≤ 1 + ε 2 t a = 0 \mathbf{a}=0 a = 0
Write λ 1 ( a ) ≥ λ 2 ( a ) ≥ ⋯ ≥ 0 \lambda_1(\mathbf{a})\geq\lambda_2(\mathbf{a})\geq\cdots\geq0 λ 1 ( a ) ≥ λ 2 ( a ) ≥ ⋯ ≥ 0
Λ m ( a ) = ( λ 1 ( a ) , … , λ m ( a ) ) . \Lambda_m(\mathbf{a})=(\lambda_1(\mathbf{a}),\ldots,\lambda_m(\mathbf{a})). Λ m ( a ) = ( λ 1 ( a ) , … , λ m ( a )) . The linear part of the Grunsky operator is determined by the real symmetric matrix
( L m ( a ) ) i j = { i j a i + j − 1 , i + j ≤ m + 1 , 0 , i + j > m + 1 , 1 ≤ i , j ≤ m . (13) (L_m(\mathbf{a}))_{ij}=
\begin{cases}
\sqrt{ij}\,a_{i+j-1}, & i+j\leq m+1,\\
0, & i+j>m+1,
\end{cases}
\qquad1\leq i,j\leq m. \tag*{(13)} ( L m ( a ) ) ij = { ij a i + j − 1 , 0 , i + j ≤ m + 1 , i + j > m + 1 , 1 ≤ i , j ≤ m . (13) Define its characteristic coefficients e k ( m ) ( a ) e_k^{(m)}(\mathbf{a}) e k ( m ) ( a )
det ( z I − L m ) = ∑ k = 0 m ( − 1 ) k e k ( m ) z m − k , e 0 ( m ) = 1 , E m = ( e 1 ( m ) , … , e m ( m ) ) . (14) \det(zI-L_m)=\sum_{k=0}^{m}(-1)^k e_k^{(m)}z^{m-k},\qquad e_0^{(m)}=1,\qquad\mathcal{E}_m=(e_1^{(m)},\ldots,e_m^{(m)}). \tag*{(14)} det ( z I − L m ) = k = 0 ∑ m ( − 1 ) k e k ( m ) z m − k , e 0 ( m ) = 1 , E m = ( e 1 ( m ) , … , e m ( m ) ) . (14) Definition 3.1. A unit vector v v v
(i) det D E m ( v ) ≠ 0 \det D\mathcal{E}_m(v)\ne0 det D E m ( v ) = 0
(ii) all eigenvalues μ j ( v ) \mu_j(v) μ j ( v ) L m ( v ) L_m(v) L m ( v )
(iii) their absolute values are pairwise distinct.
The third condition makes the signed eigenvalues simple.
Theorem 3.2 (Finite-parameter spectral uniqueness and stability). For every m ≥ 1 m \ge1 m ≥ 1 v \mathbf{v} v v \mathbf{v} v (10) . There exist ε , δ , c , C > 0 \varepsilon,\delta,c,C>0 ε , δ , c , C > 0 C = C ( v , ε , δ ) \mathcal{C}=\mathcal{C}(\mathbf{v},\varepsilon,\delta) C = C ( v , ε , δ )
(i) Each Ψ a \Psi_{\mathbf{a}} Ψ a m m m
(ii) For all a , b ∈ C ∪ { 0 } \mathbf{a},\mathbf{b}\in\mathcal{C}\cup\{\mathbf{0}\} a , b ∈ C ∪ { 0 }
c ∣ a − b ∣ ≤ ∣ Λ m ( a ) − Λ m ( b ) ∣ ≤ C ∣ a − b ∣ . (15) c\lvert\mathbf{a}-\mathbf{b}\rvert\le\lvert\Lambda_m(\mathbf{a})-\Lambda_m(\mathbf{b})\rvert\le C\lvert\mathbf{a}-\mathbf{b}\rvert. \tag*{(15)} c ∣ a − b ∣ ≤ ∣ Λ m ( a ) − Λ m ( b )∣ ≤ C ∣ a − b ∣ . (15) Thus these m m m
(iii) For all a , b ∈ C ∪ { 0 } \mathbf{a},\mathbf{b}\in\mathcal{C}\cup\{\mathbf{0}\} a , b ∈ C ∪ { 0 } c ( a ) = ( c n ( a ) ) n ≥ 1 \mathbf{c}(\mathbf{a})=(c_n(\mathbf{a}))_{n\ge1} c ( a ) = ( c n ( a ) ) n ≥ 1 (11) . Then, for each fixed s ≥ 0 s\ge0 s ≥ 0
d H ( K a , K b ) + ∥ c ( a ) − c ( b ) ∥ ℓ s 2 ≤ C s ∣ Λ m ( a ) − Λ m ( b ) ∣ , (16) d_H(K_{\mathbf{a}},K_{\mathbf{b}})+\lVert\mathbf{c}(\mathbf{a})-\mathbf{c}(\mathbf{b})\rVert_{\ell_s^2}\le C_s\lvert\Lambda_m(\mathbf{a})-\Lambda_m(\mathbf{b})\rvert, \tag*{(16)} d H ( K a , K b ) + ∥ c ( a ) − c ( b ) ∥ ℓ s 2 ≤ C s ∣ Λ m ( a ) − Λ m ( b )∣ , (16) where ∥ c ∥ ℓ s 2 2 = ∑ n ≥ 1 n 2 s ∣ c n ∣ 2 \lVert\mathbf{c}\rVert_{\ell_s^2}^2=\sum_{n\ge1}n^{2s}\lvert c_n\rvert^2 ∥ c ∥ ℓ s 2 2 = ∑ n ≥ 1 n 2 s ∣ c n ∣ 2
The constants may depend on m m m F F F v \mathbf{v} v a , b → 0 \mathbf{a},\mathbf{b}\to0 a , b → 0 m m m
The proof begins with geometric coordinates and estimates for the full Grunsky operator. We then construct admissible directions for its finite-dimensional linearisation and compare the derivatives of the two spectral maps uniformly near the disk. Integrating this comparison along segments in a convex cone gives (15) . Uniform control of the differential throughout the cone is essential to this final step.
Conformal geometry and geometric coordinates
Define
T n ( w , z ) = ∑ k = 1 n w − k z − ( n + 1 − k ) . T_n(w,z)=\sum_{k=1}^{n}w^{-k}z^{-(n+1-k)}. T n ( w , z ) = k = 1 ∑ n w − k z − ( n + 1 − k ) . The divided difference of (11) is
Ψ a ( w ) − Ψ a ( z ) w − z = 1 − S a ( w , z ) , S a = ∑ n = 1 m a n T n + ∣ a ∣ 2 ∑ n > m f n T n . (17) \frac{\Psi_{\mathbf{a}}(w)-\Psi_{\mathbf{a}}(z)}{w-z}=1-S_{\mathbf{a}}(w,z),\qquad S_{\mathbf{a}}=\sum_{n=1}^{m}a_nT_n+\lvert\mathbf{a}\rvert^2\sum_{n>m}f_nT_n. \tag*{(17)} w − z Ψ a ( w ) − Ψ a ( z ) = 1 − S a ( w , z ) , S a = n = 1 ∑ m a n T n + ∣ a ∣ 2 n > m ∑ f n T n . (17) Since ∣ T n ( w , z ) ∣ ≤ n \lvert T_n(w,z)\rvert\le n ∣ T n ( w , z )∣ ≤ n ∣ w ∣ , ∣ z ∣ ≥ 1 \lvert w\rvert,\lvert z\rvert\ge1 ∣ w ∣ , ∣ z ∣ ≥ 1
∑ n = 1 m n ∣ a n ∣ + ∣ a ∣ 2 ∑ n > m n ∣ f n ∣ < 1 (18) \sum_{n=1}^{m}n\lvert a_n\rvert+\lvert\mathbf{a}\rvert^2\sum_{n>m}n\lvert f_n\rvert<1 \tag*{(18)} n = 1 ∑ m n ∣ a n ∣ + ∣ a ∣ 2 n > m ∑ n ∣ f n ∣ < 1 (18) ensures injectivity on the closed exterior disk. Taking z → w z\to w z → w Ψ a ′ ( w ) ≠ 0 \Psi_{\mathbf{a}}'(w)\ne0 Ψ a ′ ( w ) = 0 Ψ a ( w ) = w + O ( w − 1 ) \Psi_{\mathbf{a}}(w)=w+O(w^{-1}) Ψ a ( w ) = w + O ( w − 1 ) Ψ a \Psi_{\mathbf{a}} Ψ a
Exponential decay of the tail makes z a ( θ ) = Ψ a ( e i θ ) z_{\mathbf{a}}(\theta)=\Psi_{\mathbf{a}}(e^{i\theta}) z a ( θ ) = Ψ a ( e i θ ) θ \theta θ z a → e i θ z_{\mathbf{a}}\to e^{i\theta} z a → e i θ C k C^k C k k k k
Im ( z a ′ ‾ z a ′ ′ ) ∣ z a ′ ∣ 3 ⟶ 1 \frac{\operatorname{Im}(\overline{z_{\mathbf{a}}'}z_{\mathbf{a}}'')}{\lvert z_{\mathbf{a}}'\rvert^3}\longrightarrow1 ∣ z a ′ ∣ 3 Im ( z a ′ z a ′′ ) ⟶ 1 uniformly in θ \theta θ
Near the disk, the Laurent parameters give quantitative geometric coordinates. To compare parameter distance with Hausdorff distance on a small real parameter ball, we differentiate the support function.
Proposition 3.3 (Uniform geometric coordinates). There exist r 0 > 0 r_0 > 0 r 0 > 0 C + > 0 C_+ > 0 C + > 0 ∣ a ∣ , ∣ b ∣ ≤ r 0 \lvert\mathbf{a}\rvert,\lvert\mathbf{b}\rvert\le r_0 ∣ a ∣ , ∣ b ∣ ≤ r 0
1 2 2 ∣ a − b ∣ ≤ d H ( K a , K b ) ≤ C + ∣ a − b ∣ . (19) \frac{1}{2\sqrt{2}}\lvert\mathbf{a}-\mathbf{b}\rvert\le d_H(K_{\mathbf{a}},K_{\mathbf{b}}) \le C_+\lvert\mathbf{a}-\mathbf{b}\rvert. \tag*{(19)} 2 2 1 ∣ a − b ∣ ≤ d H ( K a , K b ) ≤ C + ∣ a − b ∣ . (19) For every fixed s ≥ 0 s \ge0 s ≥ 0 F s 2 = ∑ n > m n 2 s ∣ f n ∣ 2 F_s^2 = \sum_{n>m} n^{2s}\lvert f_n\rvert^2 F s 2 = ∑ n > m n 2 s ∣ f n ∣ 2
∥ c ( a ) − c ( b ) ∥ ℓ s 2 ≤ m 2 s + 4 r 0 2 F s 2 ∣ a − b ∣ . (20) \lVert c(\mathbf{a})-c(\mathbf{b})\rVert_{\ell_s^2} \le\sqrt{m^{2s}+4r_0^2F_s^2}\lvert\mathbf{a}-\mathbf{b}\rvert. \tag*{(20)} ∥ c ( a ) − c ( b ) ∥ ℓ s 2 ≤ m 2 s + 4 r 0 2 F s 2 ∣ a − b ∣ . (20) Proof. For a compact convex body K K K
h K ( θ ) = max z ∈ K ℜ ( z e − i θ ) . h_K(\theta)=\max_{z\in K}\Re(ze^{-i\theta}). h K ( θ ) = z ∈ K max ℜ ( z e − i θ ) . For convex bodies, d H ( K , L ) = ∥ h K − h L ∥ ∞ d_H(K,L)=\lVert h_K-h_L\rVert_\infty d H ( K , L ) = ∥ h K − h L ∥ ∞ K ⊂ L + d D ‾ K\subset L+d\overline{D} K ⊂ L + d D h K ≤ h L + d h_K\le h_L+d h K ≤ h L + d x ∈ K x\in K x ∈ K L + d D ‾ L+d\overline{D} L + d D y y y ( x − y ) ⋅ ( z − y ) ≤ 0 (x-y)\cdot(z-y)\le0 ( x − y ) ⋅ ( z − y ) ≤ 0 z z z x − y x-y x − y x x x K K K L L L
We have h a ( θ ) = max t ℜ ( Ψ a ( e i t ) e − i θ ) h_{\mathbf{a}}(\theta)=\max_t\Re(\Psi_{\mathbf{a}}(e^{it})e^{-i\theta}) h a ( θ ) = max t ℜ ( Ψ a ( e i t ) e − i θ ) a = 0 \mathbf{a}=0 a = 0 t = θ t=\theta t = θ t t t − 1 -1 − 1
The first derivative in t t t
h 0 = 1 , ∂ a n h 0 ( θ ) = cos ( ( n + 1 ) θ ) . h_0=1,\qquad\partial_{a_n}h_0(\theta)=\cos((n+1)\theta). h 0 = 1 , ∂ a n h 0 ( θ ) = cos (( n + 1 ) θ ) . It follows that
h a = 1 + ∑ n = 1 m a n cos ( ( n + 1 ) θ ) + R a , ∥ R a − R b ∥ ∞ ≤ M r 0 ∣ a − b ∣ . h_{\mathbf{a}}=1+\sum_{n=1}^{m}a_n\cos((n+1)\theta)+\mathcal{R}_{\mathbf{a}},\qquad\lVert\mathcal{R}_{\mathbf{a}}-\mathcal{R}_{\mathbf{b}}\rVert_\infty\le Mr_0\lvert\mathbf{a}-\mathbf{b}\rvert. h a = 1 + n = 1 ∑ m a n cos (( n + 1 ) θ ) + R a , ∥ R a − R b ∥ ∞ ≤ M r 0 ∣ a − b ∣ . For the remainder bound, bounded second parameter derivatives imply ∥ D R a ∥ ≤ M ∣ a ∣ \lVert D\mathcal{R}_{\mathbf{a}}\rVert\le M\lvert\mathbf{a}\rvert ∥ D R a ∥ ≤ M ∣ a ∣
Orthogonality in L 2 ( d θ / ( 2 π ) ) L^2(d\theta/(2\pi)) L 2 ( d θ / ( 2 π ))
∥ ∑ n = 1 m ( a n − b n ) cos ( ( n + 1 ) θ ) ∥ L 2 = ∣ a − b ∣ 2 . \left\lVert\sum_{n=1}^{m}(a_n-b_n)\cos((n+1)\theta)\right\rVert_{L^2}=\frac{\lvert\mathbf{a}-\mathbf{b}\rvert}{\sqrt{2}}. n = 1 ∑ m ( a n − b n ) cos (( n + 1 ) θ ) L 2 = 2 ∣ a − b ∣ . The uniform norm dominates this normalised L 2 L^2 L 2
( 1 2 − M r 0 ) ∣ a − b ∣ ≤ ∥ h a − h b ∥ ∞ ≤ ( m + M r 0 ) ∣ a − b ∣ . \left(\frac{1}{\sqrt{2}}-Mr_0\right)\lvert\mathbf{a}-\mathbf{b}\rvert\le\lVert h_{\mathbf{a}}-h_{\mathbf{b}}\rVert_\infty\le(\sqrt{m}+Mr_0)\lvert\mathbf{a}-\mathbf{b}\rvert. ( 2 1 − M r 0 ) ∣ a − b ∣ ≤ ∥ h a − h b ∥ ∞ ≤ ( m + M r 0 ) ∣ a − b ∣ . Taking M r 0 ≤ 1 / ( 2 2 ) Mr_0\le1/(2\sqrt{2}) M r 0 ≤ 1/ ( 2 2 ) (19) . The coefficient difference satisfies the exact identity
∥ c ( a ) − c ( b ) ∥ ℓ s 2 2 = ∑ n = 1 m n 2 s ∣ a n − b n ∣ 2 + ( ∣ a ∣ 2 − ∣ b ∣ 2 ) 2 F s 2 . \|c(\mathbf{a})-c(\mathbf{b})\|_{\ell^2_s}^2=\sum_{n=1}^{m}n^{2s}|a_n-b_n|^2+\left(|\mathbf{a}|^2-|\mathbf{b}|^2\right)^2F_s^2. ∥ c ( a ) − c ( b ) ∥ ℓ s 2 2 = n = 1 ∑ m n 2 s ∣ a n − b n ∣ 2 + ( ∣ a ∣ 2 − ∣ b ∣ 2 ) 2 F s 2 . Using ∣ ∣ a ∣ 2 − ∣ b ∣ 2 ∣ ≤ 2 r 0 ∣ a − b ∣ ||\mathbf{a}|^2-|\mathbf{b}|^2|\le2r_0|\mathbf{a}-\mathbf{b}| ∣∣ a ∣ 2 − ∣ b ∣ 2 ∣ ≤ 2 r 0 ∣ a − b ∣ n 2 s ≤ m 2 s n^{2s}\le m^{2s} n 2 s ≤ m 2 s n ≤ m n\le m n ≤ m (20) . Exponential decay gives F s < ∞ F_s<\infty F s < ∞ s s s
Boundary points with the same conformal angle are at distance at most C ∣ a − b ∣ C|\mathbf{a}-\mathbf{b}| C ∣ a − b ∣ d H ( ∂ D a , ∂ D b ) d_H(\partial D_{\mathbf{a}},\partial D_{\mathbf{b}}) d H ( ∂ D a , ∂ D b )
The Grunsky operator and the NP spectrum
We use the convention
log Ψ a ( w ) − Ψ a ( z ) w − z = − ∑ i , j ≥ 1 b i j ( a ) w − i z − j , B i j ( a ) = i j b i j ( a ) . (21) \log\frac{\Psi_{\mathbf{a}}(w)-\Psi_{\mathbf{a}}(z)}{w-z}=-\sum_{i,j\ge1}b_{ij}(\mathbf{a})w^{-i}z^{-j},\qquad B_{ij}(\mathbf{a})=\sqrt{ij}\,b_{ij}(\mathbf{a}). \tag*{(21)} log w − z Ψ a ( w ) − Ψ a ( z ) = − i , j ≥ 1 ∑ b ij ( a ) w − i z − j , B ij ( a ) = ij b ij ( a ) . (21) Symmetry gives b i j = b j i b_{ij}=b_{ji} b ij = b j i (21) are logarithmic Grunsky coefficients; the corresponding Faber-polynomial coefficients carry an additional index factor.
Lemma 3.4 (Analytic operator expansion). On a sufficiently small parameter ball, B ( a ) B(\mathbf{a}) B ( a ) ℓ 2 ( N ) \ell^2(\mathbb{N}) ℓ 2 ( N ) a \mathbf{a} a
B ( a ) = L ~ m ( a ) + R ( a ) , ∥ R ( a ) ∥ H S ≤ C ∣ a ∣ 2 , ∥ D R ( a ) ∥ R m → H S ≤ C ∣ a ∣ , (22) B(\mathbf{a})=\widetilde{L}_m(\mathbf{a})+R(\mathbf{a}),\qquad\|R(\mathbf{a})\|_{\mathrm{HS}}\le C|\mathbf{a}|^2,\qquad\|DR(\mathbf{a})\|_{\mathbb{R}^m\to\mathrm{HS}}\le C|\mathbf{a}|, \tag*{(22)} B ( a ) = L m ( a ) + R ( a ) , ∥ R ( a ) ∥ HS ≤ C ∣ a ∣ 2 , ∥ D R ( a ) ∥ R m → HS ≤ C ∣ a ∣ , (22) where L ~ m = L m ⊕ 0 \widetilde{L}_m=L_m\oplus0 L m = L m ⊕ 0
Proof. Choose r ∗ < r < 1 r_*<r<1 r ∗ < r < 1 ∣ a ∣ 2 |\mathbf{a}|^2 ∣ a ∣ 2 ∑ a n 2 \sum a_n^2 ∑ a n 2 ∣ w ∣ , ∣ z ∣ ≥ r |w|,|z|\ge r ∣ w ∣ , ∣ z ∣ ≥ r
∣ T n ( w , z ) ∣ ≤ n r − n − 1 , ∑ n > m n ∣ f n ∣ r − n − 1 < ∞ . |T_n(w,z)|\le nr^{-n-1},\qquad\sum_{n>m}n|f_n|r^{-n-1}<\infty. ∣ T n ( w , z ) ∣ ≤ n r − n − 1 , n > m ∑ n ∣ f n ∣ r − n − 1 < ∞. On a sufficiently small complex parameter ball, we have ∣ S a ∣ < q < 1 |\mathcal{S}_{\mathbf{a}}|<q<1 ∣ S a ∣ < q < 1
− log ( 1 − S a ) = ∑ k ≥ 1 S a k k . (23) -\log(1-\mathcal{S}_{\mathbf{a}})=\sum_{k\ge1}\frac{\mathcal{S}_{\mathbf{a}}^k}{k}. \tag*{(23)} − log ( 1 − S a ) = k ≥ 1 ∑ k S a k . (23) converges normally. Cauchy estimates in w − 1 , z − 1 w^{-1},z^{-1} w − 1 , z − 1
∣ b i j ( a ) ∣ ≤ C r i + j . |b_{ij}(\mathbf{a})|\le Cr^{i+j}. ∣ b ij ( a ) ∣ ≤ C r i + j . The convergence of ∑ i , j i j r 2 i + 2 j \sum_{i,j}ijr^{2i+2j} ∑ i , j ij r 2 i + 2 j
The degree-one term of (23) is ∑ n ≤ m a n T n \sum_{n\le m}a_nT_n ∑ n ≤ m a n T n w − i z − j w^{-i}z^{-j} w − i z − j a i + j − 1 a_{i+j-1} a i + j − 1 i + j ≤ m + 1 i+j\le m+1 i + j ≤ m + 1 i j \sqrt{ij} ij (13) . All remaining terms have parameter degree at least two. Taylor’s theorem, with a bound on the second derivative of the Hilbert–Schmidt-valued map, gives both estimates in (22) . For real parameters, the operator is real and symmetric, hence self-adjoint.
For real coefficients, the NP–Grunsky representation gives the bounded similarity
K D a ∗ ∣ H 0 − 1 / 2 ∼ 1 2 ( 0 B ( a ) B ( a ) 0 ) . (24) K^{*}_{D_a}\big|_{H^{-1/2}_0}\sim\frac{1}{2}\begin{pmatrix} 0 & B(\mathbf{a}) \\ B(\mathbf{a}) & 0 \end{pmatrix}. \tag*{(24)} K D a ∗ H 0 − 1/2 ∼ 2 1 ( 0 B ( a ) B ( a ) 0 ) . (24) To describe the conformal density coordinates used here, set
ζ ± n ( Ψ ( e i θ ) ) = n e ± i n θ h ( θ ) , h ( θ ) = ∣ ∂ θ Ψ ( e i θ ) ∣ . \zeta_{\pm n}(\Psi(e^{i\theta}))=\frac{\sqrt{n}e^{\pm in\theta}}{h(\theta)},\qquad h(\theta)=\left|\partial_\theta\Psi(e^{i\theta})\right|. ζ ± n ( Ψ ( e i θ )) = h ( θ ) n e ± in θ , h ( θ ) = ∂ θ Ψ ( e i θ ) . Expansion in these densities gives a bounded identification of ℓ 2 ⊕ ℓ 2 \ell^2\oplus\ell^2 ℓ 2 ⊕ ℓ 2 H − 1 / 2 H^{-1/2} H − 1/2
K D ∗ ζ n = 1 2 ∑ k ≥ 1 B k n ζ − k , K D ∗ ζ − n = 1 2 ∑ k ≥ 1 B k n ζ k . K^{*}_{D}\zeta_n=\frac{1}{2}\sum_{k\ge1}B_{kn}\zeta_{-k},\qquad K^{*}_{D}\zeta_{-n}=\frac{1}{2}\sum_{k\ge1}B_{kn}\zeta_k. K D ∗ ζ n = 2 1 k ≥ 1 ∑ B k n ζ − k , K D ∗ ζ − n = 2 1 k ≥ 1 ∑ B k n ζ k . This proves (24) ; see [12 Proposition 3.1 and equation (4.3)] and the derivation in [13 ] . The classical relation with Fredholm eigenvalues is discussed in [19 ] .
The unitary change ( x , y ) ↦ 2 − 1 / 2 ( x + y , x − y ) (x,y)\mapsto2^{-1/2}(x+y,x-y) ( x , y ) ↦ 2 − 1/2 ( x + y , x − y ) 1 2 diag ( B , − B ) \frac{1}{2}\operatorname{diag}(B,-B) 2 1 diag ( B , − B ) 1 2 \frac{1}{2} 2 1 B B B
s j ( B ( a ) ) = 2 λ j ( a ) . (25) s_j(B(\mathbf{a}))=2\lambda_j(\mathbf{a}). \tag*{(25)} s j ( B ( a )) = 2 λ j ( a ) . (25) The real Laurent coefficients ensure that B B B
Nondegenerate directions
Lemma 3.5 (A determinant identity). For every m m m
det L m ( a ) = κ m a m m , κ m = ( − 1 ) m ( m − 1 ) / 2 m ! . (26) \det L_m(\mathbf{a})=\kappa_m a_m^m,\qquad\kappa_m=(-1)^{m(m-1)/2}m!. \tag*{(26)} det L m ( a ) = κ m a m m , κ m = ( − 1 ) m ( m − 1 ) /2 m ! . (26) Proof. Let J J J m m m ( L m J ) i k = 0 (L_mJ)_{ik}=0 ( L m J ) ik = 0 i > k i>k i > k i ( m + 1 − i ) a m \sqrt{i}(m+1-i)a_m i ( m + 1 − i ) a m m ! a m m m!a_m^m m ! a m m det J = ( − 1 ) m ( m − 1 ) / 2 \det J=(-1)^{m(m-1)/2} det J = ( − 1 ) m ( m − 1 ) /2 L m L_m L m (26) .
Proposition 3.6 (Existence in the positive orthant). For every m ≥ 1 m\ge1 m ≥ 1 p ∈ ( 0 , ∞ ) m \mathbf{p}\in(0,\infty)^m p ∈ ( 0 , ∞ ) m D E m ( p ) D E_m(\mathbf{p}) D E m ( p ) L m ( p ) L_m(\mathbf{p}) L m ( p ) v = p / ∣ p ∣ \mathbf{v}=\mathbf{p}/|\mathbf{p}| v = p /∣ p ∣
Proof. We induct on m m m m = 1 m=1 m = 1 L 1 ( a 1 ) = ( a 1 ) L_1(a_1)=(a_1) L 1 ( a 1 ) = ( a 1 ) a 1 > 0 a_1>0 a 1 > 0 p ′ ∈ R m − 1 \mathbf{p}'\in\mathbb{R}^{m-1} p ′ ∈ R m − 1 a m = 0 a_m=0 a m = 0
L m ( p ′ , 0 ) = L m − 1 ( p ′ ) ⊕ 0 , e k ( m ) ( p ′ , 0 ) = e k ( m − 1 ) ( p ′ ) ( 1 ≤ k < m ) . L_m(\mathbf{p}',0)=L_{m-1}(\mathbf{p}')\oplus0,\qquad e_k^{(m)}(\mathbf{p}',0)=e_k^{(m-1)}(\mathbf{p}')\quad(1\le k<m). L m ( p ′ , 0 ) = L m − 1 ( p ′ ) ⊕ 0 , e k ( m ) ( p ′ , 0 ) = e k ( m − 1 ) ( p ′ ) ( 1 ≤ k < m ) . The last characteristic coefficient is the determinant in (26) . Expanding the Jacobian determinant along its last row gives
det D E m ( a ) = m κ m a m m − 1 det [ ∂ e i ( m ) ∂ a j ( a ) ] i , j = 1 m − 1 . (27) \det D E_m(\mathbf{a})=m\kappa_m a_m^{m-1}\det\left[\frac{\partial e_i^{(m)}}{\partial a_j}(\mathbf{a})\right]_{i,j=1}^{m-1}. \tag*{(27)} det D E m ( a ) = m κ m a m m − 1 det [ ∂ a j ∂ e i ( m ) ( a ) ] i , j = 1 m − 1 . (27) At ( p ′ , 0 ) (\mathbf{p}',0) ( p ′ , 0 ) det D E m − 1 ( p ′ ) ≠ 0 \det D E_{m-1}(\mathbf{p}')\ne0 det D E m − 1 ( p ′ ) = 0 a m a_m a m (27) is then nonzero as well.
The first m − 1 m-1 m − 1 a m → 0 a_m \to0 a m → 0 (26) makes it nonzero for a m > 0 a_m > 0 a m > 0 a m a_m a m e k ( m ) e_k^{(m)} e k ( m ) k k k det D E m ( t p ) = t m ( m − 1 ) / 2 det D E m ( p ) \det D\mathcal{E}_m(tp) = t^{m(m-1)/2}\det D\mathcal{E}_m(p) det D E m ( tp ) = t m ( m − 1 ) /2 det D E m ( p ) t > 0 t > 0 t > 0 v = p / ∣ p ∣ v = p/|p| v = p /∣ p ∣
The construction takes each new coefficient sufficiently small relative to the preceding ones. Admissibility is also open and dense in the positive orthant. The Jacobian is a nonzero polynomial. Nonzero eigenvalues with pairwise distinct absolute values correspond to positive simple eigenvalues of L m 2 L_m^2 L m 2
Lemma 3.7 (Invertible singular-value differential). At an admissible v v v
S 0 ( a ) = ( s 1 0 ( a ) , … , s m 0 ( a ) ) , s j 0 ( a ) = s j ( L m ( a ) ) S_0(\mathbf{a}) = (s_1^0(\mathbf{a}),\ldots,s_m^0(\mathbf{a})), \qquad s_j^0(\mathbf{a}) = s_j(L_m(\mathbf{a})) S 0 ( a ) = ( s 1 0 ( a ) , … , s m 0 ( a )) , s j 0 ( a ) = s j ( L m ( a )) is smooth near v v v A = D S 0 ( v ) A = DS_0(v) A = D S 0 ( v ) v v v S 0 S_0 S 0 D S 0 DS_0 D S 0
Proof. Choose smooth local labels μ j ( a ) \mu_j(\mathbf{a}) μ j ( a ) ∏ i < j ( μ i − μ j ) \prod_{i<j}(\mu_i-\mu_j) ∏ i < j ( μ i − μ j ) ∏ j ( z − μ j ) \prod_j(z-\mu_j) ∏ j ( z − μ j ) ( m − 1 ) (m-1) ( m − 1 ) D E m ( v ) D\mathcal{E}_m(v) D E m ( v ) D ( μ 1 , … , μ m ) ( v ) D(\mu_1,\ldots,\mu_m)(v) D ( μ 1 , … , μ m ) ( v )
Near v v v L m L_m L m S 0 ( t a ) = t S 0 ( a ) S_0(t\mathbf{a}) = tS_0(\mathbf{a}) S 0 ( t a ) = t S 0 ( a ) t > 0 t > 0 t > 0 D S 0 ( t a ) = D S 0 ( a ) DS_0(t\mathbf{a}) = DS_0(\mathbf{a}) D S 0 ( t a ) = D S 0 ( a )
Differentiated spectral estimates
Near the disk, the linearised eigenvalue gaps decay linearly in the parameters, while the Grunsky remainder is quadratic. This difference of orders also controls derivatives of the leading spectral branches. Analytic perturbation theory supplies smooth local eigenvectors for isolated simple eigenvalues; see [14 Chapter VII, Section 1] . We now establish estimates that remain uniform as the parameters tend to zero within the cone.
Fix an admissible v v v γ > 0 \gamma> 0 γ > 0 r = ∣ a ∣ r = |\mathbf{a}| r = ∣ a ∣
∣ μ j ∣ ≥ γ r , ∣ μ i − μ j ∣ ≥ γ r , ∣ ∣ μ i ∣ − ∣ μ j ∣ ∣ ≥ γ r ( i ≠ j ) . (28) |\mu_j| \ge\gamma r,\qquad|\mu_i-\mu_j| \ge\gamma r,\qquad\bigl||\mu_i|-|\mu_j|\bigr| \ge\gamma r\quad(i \ne j). \tag*{(28)} ∣ μ j ∣ ≥ γ r , ∣ μ i − μ j ∣ ≥ γ r , ∣ μ i ∣ − ∣ μ j ∣ ≥ γ r ( i = j ) . (28) Here μ j \mu_j μ j L ~ m \widetilde{L}_m L m
Lemma 3.8 (Eigenvalue and derivative estimates). On a sufficiently small such cone, the first m m m B B B β j \beta_j β j m m m μ j \mu_j μ j L ~ m \widetilde{L}_m L m
S ( a ) = ( s 1 ( B ( a ) ) , … , s m ( B ( a ) ) ) S(\mathbf{a}) = (s_1(B(\mathbf{a})),\ldots,s_m(B(\mathbf{a}))) S ( a ) = ( s 1 ( B ( a )) , … , s m ( B ( a ))) satisfies the uniform estimates
∣ S ( a ) − S 0 ( a ) ∣ ≤ C r 2 , ∥ D S ( a ) − D S 0 ( a ) ∥ ≤ C r . (29) \lvert S(\mathbf{a})-S_0(\mathbf{a})\rvert\leq Cr^2,\qquad\lVert DS(\mathbf{a})-DS_0(\mathbf{a})\rVert\leq Cr. \tag*{(29)} ∣ S ( a ) − S 0 ( a )∣ ≤ C r 2 , ∥ D S ( a ) − D S 0 ( a )∥ ≤ C r . (29) The remaining singular values obey s m + 1 ( B ) ≤ C r 2 s_{m+1}(B) \leq Cr^2 s m + 1 ( B ) ≤ C r 2
Proof. Consider the circles ∣ z − μ j ∣ = γ r / 4 \lvert z-\mu_j\rvert=\gamma r/4 ∣ z − μ j ∣ = γ r /4 σ ( L ~ m ) \sigma(\widetilde{L}_m) σ ( L m ) γ r / 4 \gamma r/4 γ r /4 ∥ R ∥ o p ≤ γ r / 8 \lVert R\rVert_{\mathrm{op}}\leq\gamma r/8 ∥ R ∥ op ≤ γ r /8 L ~ m + τ R \widetilde{L}_m+\tau R L m + τ R 0 ≤ τ ≤ 1 0\leq\tau\leq1 0 ≤ τ ≤ 1 β j \beta_j β j ∣ β j − μ j ∣ ≤ ∥ R ∥ o p ≤ C r 2 \lvert\beta_j-\mu_j\rvert\leq\lVert R\rVert_{\mathrm{op}}\leq Cr^2 ∣ β j − μ j ∣ ≤ ∥ R ∥ op ≤ C r 2 ∥ R ∥ o p \lVert R\rVert_{\mathrm{op}} ∥ R ∥ op
Since L ~ m \widetilde{L}_m L m m m m s m + 1 ( B ) ≤ ∥ R ∥ o p ≤ C r 2 s_{m+1}(B)\leq\lVert R\rVert_{\mathrm{op}}\leq Cr^2 s m + 1 ( B ) ≤ ∥ R ∥ op ≤ C r 2 m m m γ r − C r 2 \gamma r-Cr^2 γ r − C r 2 m m m (28) preserve the signs and absolute ordering of the eigenvalue branches.
To estimate derivatives, choose unit eigenvectors u j u_j u j v j v_j v j B B B L ~ m \widetilde{L}_m L m P j = ⟨ ⋅ , v j ⟩ v j P_j=\langle\cdot,v_j\rangle v_j P j = ⟨ ⋅ , v j ⟩ v j
( L ~ m − β j ) u j = − R u j (\widetilde{L}_m-\beta_j)u_j=-Ru_j ( L m − β j ) u j = − R u j onto v j ⊥ v_j^\perp v j ⊥ (28) , the restriction of L ~ m − β j \widetilde{L}_m-\beta_j L m − β j C / r C/r C / r
∥ ( I − P j ) u j ∥ ≤ C r − 1 ∥ R ∥ o p ≤ C r . \lVert(I-P_j)u_j\rVert\leq Cr^{-1}\lVert R\rVert_{\mathrm{op}}\leq Cr. ∥( I − P j ) u j ∥ ≤ C r − 1 ∥ R ∥ op ≤ C r . Choose the phase so that ⟨ u j , v j ⟩ ≥ 0 \langle u_j,v_j\rangle\geq0 ⟨ u j , v j ⟩ ≥ 0 ∥ u j − v j ∥ ≤ C r \lVert u_j-v_j\rVert\leq Cr ∥ u j − v j ∥ ≤ C r
For each coordinate a k a_k a k
∂ a k β j = ⟨ u j , ( ∂ a k B ) u j ⟩ , ∂ a k μ j = ⟨ v j , ( ∂ a k L ~ m ) v j ⟩ . \partial_{a_k}\beta_j=\langle u_j,(\partial_{a_k}B)u_j\rangle,\qquad\partial_{a_k}\mu_j=\langle v_j,(\partial_{a_k}\widetilde{L}_m)v_j\rangle. ∂ a k β j = ⟨ u j , ( ∂ a k B ) u j ⟩ , ∂ a k μ j = ⟨ v j , ( ∂ a k L m ) v j ⟩ . Subtract these identities and apply (22) . The matrix ∂ a k L ~ m \partial_{a_k}\widetilde{L}_m ∂ a k L m
∣ ∂ a k β j − ∂ a k μ j ∣ ≤ ∥ ∂ a k R ∥ o p + 2 ∥ ∂ a k L ~ m ∥ o p ∥ u j − v j ∥ ≤ C r . \lvert\partial_{a_k}\beta_j-\partial_{a_k}\mu_j\rvert\leq\lVert\partial_{a_k}R\rVert_{\mathrm{op}}+2\lVert\partial_{a_k}\widetilde{L}_m\rVert_{\mathrm{op}}\lVert u_j-v_j\rVert\leq Cr. ∣ ∂ a k β j − ∂ a k μ j ∣ ≤ ∥ ∂ a k R ∥ op + 2 ∥ ∂ a k L m ∥ op ∥ u j − v j ∥ ≤ C r . The signs and ordering are fixed, so the same bounds hold for the singular values. Combining the finitely many branch and coordinate estimates proves the derivative estimate in (29) .
Proof of the inverse theorem
Integrating the differential comparison along segments in the convex cone now gives the inverse Lipschitz estimate.
Proof of Theorem 3.2. Proposition 3.6 gives a positive admissible direction in every dimension. Fix any admissible v v v A = D S 0 ( v ) A=DS_0(v) A = D S 0 ( v ) α = σ min ( A ) > 0 \alpha=\sigma_{\min}(A)>0 α = σ m i n ( A ) > 0 D S 0 DS_0 D S 0
∥ D S 0 ( a ) − A ∥ ≤ α / 4 \lVert DS_0(\mathbf{a})-A\rVert\leq\alpha/4 ∥ D S 0 ( a ) − A ∥ ≤ α /4 on the punctured cone. Then choose its length δ \delta δ
∥ D S ( a ) − D S 0 ( a ) ∥ ≤ α / 4. \lVert DS(\mathbf{a})-DS_0(\mathbf{a})\rVert\leq\alpha/4. ∥ D S ( a ) − D S 0 ( a )∥ ≤ α /4. Thus ∥ D S ( a ) − A ∥ ≤ α / 2 \lVert DS(\mathbf{a})-A\rVert\le\alpha/2 ∥ D S ( a ) − A ∥ ≤ α /2 b + τ ( a − b ) \mathbf{b}+\tau(\mathbf{a}-\mathbf{b}) b + τ ( a − b )
S ( a ) − S ( b ) = A ( a − b ) + e , ∣ e ∣ ≤ α 2 ∣ a − b ∣ . S(\mathbf{a})-S(\mathbf{b})=A(\mathbf{a}-\mathbf{b})+e,\qquad|e|\le\frac{\alpha}{2}|\mathbf{a}-\mathbf{b}|. S ( a ) − S ( b ) = A ( a − b ) + e , ∣ e ∣ ≤ 2 α ∣ a − b ∣. Therefore
α 2 ∣ a − b ∣ ≤ ∣ S ( a ) − S ( b ) ∣ ≤ ( ∥ A ∥ + α / 2 ) ∣ a − b ∣ . (30) \frac{\alpha}{2}|\mathbf{a}-\mathbf{b}|\le|S(\mathbf{a})-S(\mathbf{b})|\le\left(\lVert A\rVert+\alpha/2\right)|\mathbf{a}-\mathbf{b}|. \tag*{(30)} 2 α ∣ a − b ∣ ≤ ∣ S ( a ) − S ( b ) ∣ ≤ ( ∥ A ∥ + α /2 ) ∣ a − b ∣. (30) Equation (25) yields (15) with lower constant c = α / 4 c=\alpha/4 c = α /4 S S S S 0 S_0 S 0 m m m (15) proves (16) for each fixed s ≥ 0 s\ge0 s ≥ 0
Corollary 3.9 (Prescribed analytic remainders). Fix m ≥ 1 m\ge1 m ≥ 1 0 < r ∗ < r < 1 0<r_*<r<1 0 < r ∗ < r < 1
Ψ a ( w ) = w + ∑ n = 1 m a n w − n + T ( a ; w ) , T ( a ; w ) = ∑ n > m r n ( a ) w − n . \Psi_{\mathbf{a}}(w)=w+\sum_{n=1}^{m}a_nw^{-n}+T(\mathbf{a};w),\qquad T(\mathbf{a};w)=\sum_{n>m}r_n(\mathbf{a})w^{-n}. Ψ a ( w ) = w + n = 1 ∑ m a n w − n + T ( a ; w ) , T ( a ; w ) = n > m ∑ r n ( a ) w − n . where, for some ρ > 0 \rho>0 ρ > 0 ∣ a ∣ < ρ |\mathbf{a}|<\rho ∣ a ∣ < ρ ∣ w ∣ > r ∗ |w|>r_* ∣ w ∣ > r ∗ sup ∣ w ∣ = r ∣ T ( a ; w ) ∣ ≤ C T ∣ a ∣ 2 \sup_{|w|=r}|T(\mathbf{a};w)|\le C_T|\mathbf{a}|^2 sup ∣ w ∣ = r ∣ T ( a ; w ) ∣ ≤ C T ∣ a ∣ 2 ∣ a ∣ < ρ |\mathbf{a}|<\rho ∣ a ∣ < ρ L m L_m L m c n ( a ) = a n c_n(\mathbf{a})=a_n c n ( a ) = a n n ≤ m n\le m n ≤ m c n ( a ) = r n ( a ) c_n(\mathbf{a})=r_n(\mathbf{a}) c n ( a ) = r n ( a ) n > m n>m n > m s ≥ 0 s\ge0 s ≥ 0 m m m s s s
Proof. Choose r < r 1 < 1 r<r_1<1 r < r 1 < 1 w w w
∣ r n ( a ) ∣ ≤ C ∣ a ∣ 2 r n , ∥ D r n ( a ) ∥ o p ≤ C ∣ a ∣ r n . |r_n(\mathbf{a})|\le C|\mathbf{a}|^2r^n,\qquad\lVert Dr_n(\mathbf{a})\rVert_{\mathrm{op}}\le C|\mathbf{a}|r^n. ∣ r n ( a ) ∣ ≤ C ∣ a ∣ 2 r n , ∥ D r n ( a ) ∥ op ≤ C ∣ a ∣ r n . The tail contribution to the divided difference is therefore O ( ∣ a ∣ 2 ) O(|\mathbf{a}|^2) O ( ∣ a ∣ 2 ) O ( ∣ a ∣ ) O(|\mathbf{a}|) O ( ∣ a ∣ ) ∣ w ∣ , ∣ z ∣ ≥ r 1 |w|,|z|\ge r_1 ∣ w ∣ , ∣ z ∣ ≥ r 1 L m L_m L m
These coefficient bounds also give analytic, strictly convex boundaries on a small real parameter ball. The remainder and its first parameter derivative vanish at zero, so the support-function argument in Proposition 3.3 has the same linearisation. For ∣ a ∣ , ∣ b ∣ ≤ r 0 |\mathbf{a}|,|\mathbf{b}|\le r_0 ∣ a ∣ , ∣ b ∣ ≤ r 0
( ∑ n > m n 2 s ∣ r n ( a ) − r n ( b ) ∣ 2 ) 1 / 2 ≤ C s r 0 ∣ a − b ∣ . \left(\sum_{n>m}n^{2s}|r_n(\mathbf{a})-r_n(\mathbf{b})|^2\right)^{1/2}\le C_sr_0|\mathbf{a}-\mathbf{b}|. ( n > m ∑ n 2 s ∣ r n ( a ) − r n ( b ) ∣ 2 ) 1/2 ≤ C s r 0 ∣ a − b ∣. by integration along the parameter segment and convergence of ∑ n > m n 2 s r 2 n \sum_{n>m}n^{2s}r^{2n} ∑ n > m n 2 s r 2 n L m L_m L m
Remark 3.10 (What provides global uniqueness). The estimate D S ≈ A DS\approx A D S ≈ A m = 2 m=2 m = 2
Complete spectra and further stability estimates
For the complete ordered positive spectrum, define
d s p e c , 2 ( a , b ) = ( ∑ j ≥ 1 ∣ λ j ( a ) − λ j ( b ) ∣ 2 ) 1 / 2 , d s p e c , ∞ ( a , b ) = sup j ≥ 1 ∣ λ j ( a ) − λ j ( b ) ∣ . d_{\mathrm{spec},2}(\mathbf{a},\mathbf{b})=\left(\sum_{j\geq1}\lvert\lambda_j(\mathbf{a})-\lambda_j(\mathbf{b})\rvert^2\right)^{1/2},\qquad d_{\mathrm{spec},\infty}(\mathbf{a},\mathbf{b})=\sup_{j\geq1}\lvert\lambda_j(\mathbf{a})-\lambda_j(\mathbf{b})\rvert. d spec , 2 ( a , b ) = ( j ≥ 1 ∑ ∣ λ j ( a ) − λ j ( b ) ∣ 2 ) 1/2 , d spec , ∞ ( a , b ) = j ≥ 1 sup ∣ λ j ( a ) − λ j ( b )∣ . Lemma 3.4 and (25) imply that d s p e c , 2 d_{\mathrm{spec},2} d spec , 2 ∣ Δ Λ m ∣ ≤ d s p e c , 2 \lvert\Delta\Lambda_m\rvert\leq d_{\mathrm{spec},2} ∣ Δ Λ m ∣ ≤ d spec , 2 ∣ Δ Λ m ∣ ≤ m d s p e c , ∞ \lvert\Delta\Lambda_m\rvert\leq\sqrt{m}d_{\mathrm{spec},\infty} ∣ Δ Λ m ∣ ≤ m d spec , ∞
A uniform Hilbert–Schmidt bound on the derivative of the Grunsky operator gives forward spectral stability throughout a sufficiently small parameter ball.
Lemma 3.11 (Singular-value perturbation in Hilbert–Schmidt norm). For Hilbert–Schmidt operators A , C A,C A , C
( ∑ j ≥ 1 ∣ s j ( A ) − s j ( C ) ∣ 2 ) 1 / 2 ≤ ∥ A − C ∥ H S . \left(\sum_{j\geq1}\lvert s_j(A)-s_j(C)\rvert^2\right)^{1/2}\leq\lVert A-C\rVert_{\mathrm{HS}}. ( j ≥ 1 ∑ ∣ s j ( A ) − s j ( C ) ∣ 2 ) 1/2 ≤ ∥ A − C ∥ HS . Proof. In finite dimensions, the trace inequality ℜ Tr ( A ∗ C ) ≤ ∑ j s j ( A ) s j ( C ) \Re\operatorname{Tr}(A^*C)\leq\sum_j s_j(A)s_j(C) ℜ Tr ( A ∗ C ) ≤ ∑ j s j ( A ) s j ( C )
∑ j ( s j ( A ) − s j ( C ) ) 2 ≤ ∥ A ∥ H S 2 + ∥ C ∥ H S 2 − 2 ℜ Tr ( A ∗ C ) = ∥ A − C ∥ H S 2 . \sum_j\left(s_j(A)-s_j(C)\right)^2\leq\lVert A\rVert_{\mathrm{HS}}^2+\lVert C\rVert_{\mathrm{HS}}^2-2\Re\operatorname{Tr}(A^*C)=\lVert A-C\rVert_{\mathrm{HS}}^2. j ∑ ( s j ( A ) − s j ( C ) ) 2 ≤ ∥ A ∥ HS 2 + ∥ C ∥ HS 2 − 2ℜ Tr ( A ∗ C ) = ∥ A − C ∥ HS 2 . This is the Hilbert–Schmidt case of the singular-value inequalities in [16 ] . In infinite dimensions, choose increasing finite-rank orthogonal projections P N P_N P N P N A P N → A P_NAP_N\to A P N A P N → A P N C P N → C P_NCP_N\to C P N C P N → C J J J N → ∞ N\to\infty N → ∞ J → ∞ J\to\infty J → ∞
Theorem 3.12 (Forward stability of the complete NP spectrum). On a sufficiently small real parameter ball in the fixed family, there is a constant C C C
d s p e c , 2 ( a , b ) ≤ 1 2 ∥ B ( a ) − B ( b ) ∥ H S ≤ C ∣ a − b ∣ ≤ C ′ d H ( K a , K b ) . (31) d_{\mathrm{spec},2}(\mathbf{a},\mathbf{b})\leq\frac{1}{2}\lVert B(\mathbf{a})-B(\mathbf{b})\rVert_{\mathrm{HS}}\leq C\lvert\mathbf{a}-\mathbf{b}\rvert\leq C'd_H(K_{\mathbf{a}},K_{\mathbf{b}}). \tag*{(31)} d spec , 2 ( a , b ) ≤ 2 1 ∥ B ( a ) − B ( b ) ∥ HS ≤ C ∣ a − b ∣ ≤ C ′ d H ( K a , K b ) . (31) The same bounds hold for d s p e c , ∞ d_{\mathrm{spec},\infty} d spec , ∞ ∥ c ( a ) − c ( b ) ∥ ℓ s 2 \lVert c(\mathbf{a})-c(\mathbf{b})\rVert_{\ell_s^2} ∥ c ( a ) − c ( b ) ∥ ℓ s 2 s ≥ 0 s\geq0 s ≥ 0
Proof. Lemma 3.11 and (25) give the first inequality. For the second, integrate the uniform Hilbert–Schmidt bound on D B DB D B ℓ 2 \ell^2 ℓ 2 ∣ a − b ∣ ≤ ∥ c ( a ) − c ( b ) ∥ ℓ s 2 \lvert\mathbf{a}-\mathbf{b}\rvert\leq\lVert c(\mathbf{a})-c(\mathbf{b})\rVert_{\ell_s^2} ∣ a − b ∣ ≤ ∥ c ( a ) − c ( b ) ∥ ℓ s 2 s ≥ 0 s\geq0 s ≥ 0
Corollary 3.13 (Equivalence of geometric and spectral distances). On a cone from Theorem 3.2, including the vertex, the following distances are mutually comparable for each fixed s ≥ 0 s\geq0 s ≥ 0
∣ a − b ∣ , ∥ c ( a ) − c ( b ) ∥ ℓ s 2 , d H ( K a , K b ) , ∣ Λ m ( a ) − Λ m ( b ) ∣ , d s p e c , 2 , d s p e c , ∞ . \lvert\mathbf{a}-\mathbf{b}\rvert,\qquad\lVert c(\mathbf{a})-c(\mathbf{b})\rVert_{\ell_s^2},\qquad d_H(K_{\mathbf{a}},K_{\mathbf{b}}),\qquad\lvert\Lambda_m(\mathbf{a})-\Lambda_m(\mathbf{b})\rvert,\qquad d_{\mathrm{spec},2},\qquad d_{\mathrm{spec},\infty}. ∣ a − b ∣ , ∥ c ( a ) − c ( b ) ∥ ℓ s 2 , d H ( K a , K b ) , ∣ Λ m ( a ) − Λ m ( b )∣ , d spec , 2 , d spec , ∞ . The comparison constants may depend on m m m s s s F F F
Proof. Theorem 3.2 , Proposition 3.3 , and Theorem 3.12 give the comparisons, together with ∣ Δ Λ m ∣ ≤ d s p e c , 2 |\Delta\Lambda_m| \le d_{\mathrm{spec},2} ∣Δ Λ m ∣ ≤ d spec , 2 ∣ Δ Λ m ∣ ≤ m d s p e c , ∞ |\Delta\Lambda_m| \le\sqrt{m}\,d_{\mathrm{spec},\infty} ∣Δ Λ m ∣ ≤ m d spec , ∞
The boundary comparison in Subsection 3.2 adds d H ( ∂ D a , ∂ D b ) d_H(\partial D_a,\partial D_b) d H ( ∂ D a , ∂ D b )
Table 1.
The geometric estimates and forward spectral stability hold throughout a small parameter ball. The inverse recovery result uses the nondegeneracy and angular restriction of Theorem 3.2 .
Two and three Laurent coefficients
The leading spectral maps can be inverted explicitly for two and three coefficients. The formulas give a full positive sector in dimension two and a cone about an explicit direction in dimension three.
Two coefficients
Take the two-coefficient family
Ψ a , b ( w ) = w + a w + b w 2 + ( a 2 + b 2 ) F ( w ) , F ( w ) = O ( w − 3 ) . \Psi_{a,b}(w)=w+\frac{a}{w}+\frac{b}{w^2}+(a^2+b^2)F(w),\qquad F(w)=O(w^{-3}). Ψ a , b ( w ) = w + w a + w 2 b + ( a 2 + b 2 ) F ( w ) , F ( w ) = O ( w − 3 ) . The leading matrix, its characteristic coefficients and their Jacobian are
L 2 = ( a 2 b 2 b 0 ) , e 1 = a , e 2 = − 2 b 2 , det D ( e 1 , e 2 ) = − 4 b . L_2=\begin{pmatrix} a & \sqrt{2}b \\ \sqrt{2}b & 0 \end{pmatrix},\qquad e_1=a,\qquad e_2=-2b^2,\qquad\det D(e_1,e_2)=-4b. L 2 = ( a 2 b 2 b 0 ) , e 1 = a , e 2 = − 2 b 2 , det D ( e 1 , e 2 ) = − 4 b . For a , b > 0 a,b>0 a , b > 0 d = a 2 + 8 b 2 d=\sqrt{a^2+8b^2} d = a 2 + 8 b 2
μ + = a + d 2 , μ − = a − d 2 , s 1 0 = d + a 2 , s 2 0 = d − a 2 . \mu_+=\frac{a+d}{2},\qquad\mu_-=\frac{a-d}{2},\qquad s_1^0=\frac{d+a}{2},\qquad s_2^0=\frac{d-a}{2}. μ + = 2 a + d , μ − = 2 a − d , s 1 0 = 2 d + a , s 2 0 = 2 d − a . The eigenvalues are nonzero, with absolute-value gap s 1 0 − s 2 0 = a > 0 s_1^0-s_2^0=a>0 s 1 0 − s 2 0 = a > 0
Theorem 4.1 (Uniform sector theorem for m = 2 m = 2 m = 2 0 < η < 1 0 < \eta< 1 0 < η < 1 F F F δ > 0 \delta> 0 δ > 0
Q η , δ = { ( a , b ) : 0 < a , b < δ , η ≤ b / a ≤ η − 1 } Q_{\eta,\delta} = \{(a,b): 0 < a,b < \delta,\ \eta\le b/a \le\eta^{-1}\} Q η , δ = {( a , b ) : 0 < a , b < δ , η ≤ b / a ≤ η − 1 } the map ( a , b ) ↦ ( λ 1 , λ 2 ) (a,b) \mapsto(\lambda_1,\lambda_2) ( a , b ) ↦ ( λ 1 , λ 2 ) Theorem 3.2 hold on the same sector.
Proof. The directions in this sector form a compact subset of the positive unit quadrant. Thus (28) and Lemma 3.8 hold uniformly. To compare two arbitrary points in the sector, use the inverse of the leading spectral map:
a = x − y , b = x y / 2 , ( x , y ) = ( s 1 0 , s 2 0 ) . (32) a = x-y,\qquad b = \sqrt{xy/2},\qquad(x,y) = (s_1^0,s_2^0). \tag*{(32)} a = x − y , b = x y /2 , ( x , y ) = ( s 1 0 , s 2 0 ) . (32) The sector condition implies
1 ≤ x / y ≤ M η , M η = 1 + 8 η 2 + 1 1 + 8 η 2 − 1 . 1 \le x/y \le M_\eta,\qquad M_\eta= \frac{\sqrt{1+8\eta^2}+1}{\sqrt{1+8\eta^2}-1}. 1 ≤ x / y ≤ M η , M η = 1 + 8 η 2 − 1 1 + 8 η 2 + 1 . The wedge { y > 0 , y ≤ x ≤ M η y } \{y > 0,\ y \le x \le M_\eta y\} { y > 0 , y ≤ x ≤ M η y } (32) are ( 2 2 ) − 1 y / x (2\sqrt{2})^{-1}\sqrt{y/x} ( 2 2 ) − 1 y / x ( 2 2 ) − 1 x / y (2\sqrt{2})^{-1}\sqrt{x/y} ( 2 2 ) − 1 x / y K η K_\eta K η
∣ p − q ∣ ≤ K η ∣ S 0 ( p ) − S 0 ( q ) ∣ ( p , q ∈ Q η , δ ) . |p-q| \le K_\eta|S_0(p)-S_0(q)|\qquad(p,q \in Q_{\eta,\delta}). ∣ p − q ∣ ≤ K η ∣ S 0 ( p ) − S 0 ( q ) ∣ ( p , q ∈ Q η , δ ) . Let E = S − S 0 E=S-S_0 E = S − S 0 (29) , integrated on the convex parameter region, gives ∣ E ( p ) − E ( q ) ∣ ≤ C δ ∣ p − q ∣ |E(p)-E(q)| \le C\delta|p-q| ∣ E ( p ) − E ( q ) ∣ ≤ C δ ∣ p − q ∣
∣ S ( p ) − S ( q ) ∣ ≥ ( K η − 1 − C δ ) ∣ p − q ∣ . |S(p)-S(q)| \ge(K_\eta^{-1}-C\delta)|p-q|. ∣ S ( p ) − S ( q ) ∣ ≥ ( K η − 1 − C δ ) ∣ p − q ∣. Choose δ \delta δ C δ ≤ ( 2 K η ) − 1 C\delta\le(2K_\eta)^{-1} C δ ≤ ( 2 K η ) − 1 D S 0 DS_0 D S 0 (29) give the upper bound. Equation (25) transfers both bounds to the NP eigenvalues, with a factor of two. Proposition 3.3 gives the geometric and coefficient estimates. □
Leading reconstruction and an infinite-tail example
The explicit inverse suggests the reconstruction formulas
a l e a d = 2 ( λ 1 − λ 2 ) , b l e a d = 2 λ 1 λ 2 . (33) a_{\mathrm{lead}} = 2(\lambda_1-\lambda_2),\qquad b_{\mathrm{lead}} = \sqrt{2\lambda_1\lambda_2}. \tag*{(33)} a lead = 2 ( λ 1 − λ 2 ) , b lead = 2 λ 1 λ 2 . (33) They satisfy ( a l e a d , b l e a d ) = ( a , b ) + O ( a 2 + b 2 ) (a_{\mathrm{lead}},b_{\mathrm{lead}})=(a,b)+O(a^2+b^2) ( a lead , b lead ) = ( a , b ) + O ( a 2 + b 2 ) S − S 0 = O ( ∣ p ∣ 2 ) S-S_0=O(|p|^2) S − S 0 = O ( ∣ p ∣ 2 ) (32) has a bounded derivative on a slightly enlarged positive wedge containing both spectral vectors. Thus the leading reconstruction is accurate to second order. For an example with infinitely many coefficients, take
F 2 ( w ) = e 1 / w − 1 − w − 1 − 1 2 w − 2 = ∑ n ≥ 3 w − n n ! . F_2(w)=e^{1/w}-1-w^{-1}-\frac{1}{2}w^{-2}=\sum_{n\ge3}\frac{w^{-n}}{n!}. F 2 ( w ) = e 1/ w − 1 − w − 1 − 2 1 w − 2 = n ≥ 3 ∑ n ! w − n . On the sector 1 / 2 ≤ b / a ≤ 2 1/2 \le b/a \le2 1/2 ≤ b / a ≤ 2 (18) becomes
a + 2 b + ( e − 2 ) ( a 2 + b 2 ) < 1. a+2b+(e-2)(a^2+b^2)<1. a + 2 b + ( e − 2 ) ( a 2 + b 2 ) < 1. The inverse theorem may require a smaller parameter bound. The weighted coefficient estimate has F 2 2 = ∑ n ≥ 3 n 2 s / ( n ! ) 2 F_2^2=\sum_{n\ge3} n^{2s}/(n!)^2 F 2 2 = ∑ n ≥ 3 n 2 s / ( n ! ) 2 Figure 2 shows domains in this family.
Figure 2. Two-coefficient domains with the exponential tail F 2 F_2 F 2 a = b = t a=b=t a = b = t t = 0.02 , 0.04 , 0.06 t=0.02, 0.04, 0.06 t = 0.02 , 0.04 , 0.06 100 ( ∣ Ψ ( e i θ ) ∣ − 1 ) 100(|\Psi(e^{i\theta})|-1) 100 ( ∣Ψ ( e i θ ) ∣ − 1 )
Three coefficients
Now take three coefficients:
Ψ a , b , c ( w ) = w + a w + b w 2 + c w 3 + ( a 2 + b 2 + c 2 ) F ( w ) , F ( w ) = O ( w − 4 ) . \Psi_{a,b,c}(w)=w+\frac{a}{w}+\frac{b}{w^2}+\frac{c}{w^3}+(a^2+b^2+c^2)F(w),\qquad F(w)=O(w^{-4}). Ψ a , b , c ( w ) = w + w a + w 2 b + w 3 c + ( a 2 + b 2 + c 2 ) F ( w ) , F ( w ) = O ( w − 4 ) . Its leading matrix is
L 3 ( a , b , c ) = ( a 2 b 3 c 2 b 2 c 0 3 c 0 0 ) . (34) L_3(a,b,c)=
\begin{pmatrix}
a & \sqrt{2}b & \sqrt{3}c\\
\sqrt{2}b & 2c & 0\\
\sqrt{3}c & 0 & 0
\end{pmatrix}. \tag*{(34)} L 3 ( a , b , c ) = a 2 b 3 c 2 b 2 c 0 3 c 0 0 . (34) Let T = Tr L 3 T=\operatorname{Tr}L_3 T = Tr L 3 Q = Tr L 3 2 Q=\operatorname{Tr}L_3^2 Q = Tr L 3 2 Δ 3 = det L 3 \Delta_3=\det L_3 Δ 3 = det L 3
T = a + 2 c , Q = a 2 + 4 b 2 + 10 c 2 , Δ 3 = − 6 c 3 . (35) T=a+2c,\qquad Q=a^2+4b^2+10c^2,\qquad\Delta_3=-6c^3. \tag*{(35)} T = a + 2 c , Q = a 2 + 4 b 2 + 10 c 2 , Δ 3 = − 6 c 3 . (35) The Jacobian is
det ∂ ( T , Q , Δ 3 ) ∂ ( a , b , c ) = det ( 1 0 2 2 a 8 b 20 c 0 0 − 18 c 2 ) = − 144 b c 2 . (36) \det\frac{\partial(T,Q,\Delta_3)}{\partial(a,b,c)}
=\det
\begin{pmatrix}
1 & 0 & 2\\
2a & 8b & 20c\\
0 & 0 & -18c^2
\end{pmatrix}
=-144bc^2. \tag*{(36)} det ∂ ( a , b , c ) ∂ ( T , Q , Δ 3 ) = det 1 2 a 0 0 8 b 0 2 20 c − 18 c 2 = − 144 b c 2 . (36) The change of variables ( e 1 , e 2 , e 3 ) = ( T , ( T 2 − Q ) / 2 , Δ 3 ) (e_1,e_2,e_3)=(T,(T^2-Q)/2,\Delta_3) ( e 1 , e 2 , e 3 ) = ( T , ( T 2 − Q ) /2 , Δ 3 ) − 1 / 2 -1/2 − 1/2 det D E 3 = 72 b c 2 \det DE_3=72bc^2 det D E 3 = 72 b c 2
At ( a , b , c ) = ( 1 , 1 , 1 ) (a,b,c)=(1,1,1) ( a , b , c ) = ( 1 , 1 , 1 )
p 3 ( z ) = z 3 − 3 z 2 − 3 z + 6. p_3(z)=z^3-3z^2-3z+6. p 3 ( z ) = z 3 − 3 z 2 − 3 z + 6. Evaluate it at six rational points:
Table 2.
The sign changes give a root in each of ( − 8 / 5 , − 3 / 2 ) (-8/5,-3/2) ( − 8/5 , − 3/2 ) ( 1 , 13 / 10 ) (1,13/10) ( 1 , 13/10 ) ( 3 , 7 / 2 ) (3,7/2) ( 3 , 7/2 ) (36) , the direction v = ( 1 , 1 , 1 ) / 3 v=(1,1,1)/\sqrt{3} v = ( 1 , 1 , 1 ) / 3
Corollary 4.2 (Three-eigenvalue recovery). Fix a real exponentially decaying tail beginning at w − 4 w^{-4} w − 4 ε , δ > 0 \varepsilon,\delta>0 ε , δ > 0 C ( ( 1 , 1 , 1 ) / 3 , ε , δ ) \mathcal{C}((1,1,1)/\sqrt{3},\varepsilon,\delta) C (( 1 , 1 , 1 ) / 3 , ε , δ ) ( a , b , c ) (a,b,c) ( a , b , c )
∣ ( a , b , c ) − ( a ~ , b ~ , c ~ ) ∣ ≤ C ( ∑ j = 1 3 ∣ λ j − λ ~ j ∣ 2 ) 1 / 2 . \lvert(a,b,c)-(\widetilde{a},\widetilde{b},\widetilde{c})\rvert\le C\left(\sum_{j=1}^{3}\lvert\lambda_j-\widetilde{\lambda}_j\rvert^2\right)^{1/2}. ∣( a , b , c ) − ( a , b , c )∣ ≤ C ( j = 1 ∑ 3 ∣ λ j − λ j ∣ 2 ) 1/2 . The geometric and weighted coefficient estimates of Theorem 3.2 also hold, uniformly down to the disk.
Proof. Apply Theorem 3.2 , whose hypotheses we have just checked, and narrow the cone to keep all coordinates positive.
Signed ordering and leading reconstruction
At ( 1 , 1 , 1 ) (1,1,1) ( 1 , 1 , 1 ) s 1 0 , − s 2 0 , s 3 0 s_1^0,-s_2^0,s_3^0 s 1 0 , − s 2 0 , s 3 0 x j = s j 0 x_j=s_j^0 x j = s j 0 (35) explicitly:
c = ( x 1 x 2 x 3 / 6 ) 1 / 3 , a = x 1 − x 2 + x 3 − 2 c , b = 1 2 x 1 2 + x 2 2 + x 3 2 − a 2 − 10 c 2 . (37) \begin{aligned}
c&=(x_1x_2x_3/6)^{1/3},\\
a&=x_1-x_2+x_3-2c,\\
b&=\frac{1}{2}\sqrt{x_1^2+x_2^2+x_3^2-a^2-10c^2}. \tag*{(37)}
\end{aligned} c a b = ( x 1 x 2 x 3 /6 ) 1/3 , = x 1 − x 2 + x 3 − 2 c , = 2 1 x 1 2 + x 2 2 + x 3 2 − a 2 − 10 c 2 . (37) Positivity in the prescribed cone selects the positive root for b b b ( a , b , c ) (a,b,c) ( a , b , c ) L 3 L_3 L 3 x j = 2 λ j x_j=2\lambda_j x j = 2 λ j O ( a 2 + b 2 + c 2 ) O(a^2+b^2+c^2) O ( a 2 + b 2 + c 2 ) (37) is smooth and homogeneous of degree one near the leading spectral ray, with uniformly bounded differential. After division by ∣ ( a , b , c ) ∣ \lvert(a,b,c)\rvert ∣( a , b , c )∣ (29) bounds the difference between the two spectral vectors by O ( ∣ ( a , b , c ) ∣ ) O(\lvert(a,b,c)\rvert) O (∣( a , b , c )∣)
The corresponding exponential tail is
F 3 ( w ) = e 1 / w − 1 − w − 1 − 1 2 w − 2 − 1 6 w − 3 = ∑ n ≥ 4 w − n n ! . F_3(w)=e^{1/w}-1-w^{-1}-\frac{1}{2}w^{-2}-\frac{1}{6}w^{-3}=\sum_{n\geq4}\frac{w^{-n}}{n!}. F 3 ( w ) = e 1/ w − 1 − w − 1 − 2 1 w − 2 − 6 1 w − 3 = n ≥ 4 ∑ n ! w − n . For positive coefficients, (18) becomes
a + 2 b + 3 c + ( e − 5 2 ) ( a 2 + b 2 + c 2 ) < 1. a+2b+3c+\left(e-\frac{5}{2}\right)(a^2+b^2+c^2)<1. a + 2 b + 3 c + ( e − 2 5 ) ( a 2 + b 2 + c 2 ) < 1. The inverse theorem may require a smaller radius. The weighted coefficient estimate uses F s 2 = ∑ n ≥ 4 n 2 s / ( n ! ) 2 F_s^2=\sum_{n\geq4}n^{2s}/(n!)^2 F s 2 = ∑ n ≥ 4 n 2 s / ( n ! ) 2 Figure 3 shows domains along the admissible ray.
Figure 3. Three-coefficient domains with the exponential tail F 3 F_3 F 3 a = b = c = t a=b=c=t a = b = c = t t = 0.01 , 0.02 , 0.03 t=0.01,0.02,0.03 t = 0.01 , 0.02 , 0.03 Figure 2 . All displayed curves are strictly convex by Subsection 4.3. Corollary 4.2 holds on a sufficiently small interval of the ray.
Convexity of the displayed examples
Here is a direct convexity check for the curves in Figure 2 –Figure 3 . Let q 1 = ∑ n ∣ c n ∣ q_1=\sum n|c_n| q 1 = ∑ n ∣ c n ∣ q 2 = ∑ n 2 ∣ c n ∣ q_2=\sum n^2|c_n| q 2 = ∑ n 2 ∣ c n ∣ q 1 < 1 q_1<1 q 1 < 1
Im ( z ′ ‾ z ′ ′ ) ≥ 1 − q 1 − q 2 − q 1 q 2 \operatorname{Im}(\overline{z'}z'') \ge1-q_1-q_2-q_1q_2 Im ( z ′ z ′′ ) ≥ 1 − q 1 − q 2 − q 1 q 2 for z ( θ ) = Ψ ( e i θ ) z(\theta)=\Psi(e^{i\theta}) z ( θ ) = Ψ ( e i θ ) t t t 0.43 0.43 0.43 m = 2 m=2 m = 2 0.32 0.32 0.32 m = 3 m=3 m = 3 t t t
Concluding remarks
The NP spectrum can fail to distinguish domains even when their boundaries are analytic, strictly convex and arbitrarily close to the circle, and even after allowing extended Möbius transformations. Within prescribed finite-dimensional conformal families, however, finitely many eigenvalues give Lipschitz stable recovery on suitable truncated cones, with constants uniform down to the disk.
To recover finer boundary features as m m m § 8 raises another question: which additional spectral observables separate the two branches of the moment map?
For recovery from resonance measurements, we must also extract the eigenvalues stably and control their excitation and observation weights. This is where the spectral estimates meet the polarization data of Appendix § 9 .
The two-moment map
Uniqueness on a positive sector
For m = 2 m=2 m = 2
I 1 = 4 ∑ j λ j 2 = Tr B 2 , I 2 = 16 ∑ j λ j 4 = Tr B 4 . I_1=4\sum_j\lambda_j^2=\operatorname{Tr}B^2,\qquad I_2=16\sum_j\lambda_j^4=\operatorname{Tr}B^4. I 1 = 4 j ∑ λ j 2 = Tr B 2 , I 2 = 16 j ∑ λ j 4 = Tr B 4 . Lemma 3.4 shows that these moments are real analytic in the parameters. Writing ρ = ∣ a ∣ + ∣ b ∣ \rho=|a|+|b| ρ = ∣ a ∣ + ∣ b ∣
B = L + Q + O H S ( ρ 3 ) , L = ( a 2 b 2 b 0 ) ⊕ 0 , (38) B=L+Q+O_{\mathrm{HS}}(\rho^3),\qquad L=\begin{pmatrix}a&\sqrt{2}b\\ \sqrt{2}b&0\end{pmatrix}\oplus0, \tag*{(38)} B = L + Q + O HS ( ρ 3 ) , L = ( a 2 b 2 b 0 ) ⊕ 0 , (38) where Q Q Q w − 3 w^{-3} w − 3 L L L i + j ≤ 3 i+j\leq3 i + j ≤ 3 Q Q Q i + j ≥ 4 i+j\geq4 i + j ≥ 4 Tr ( L Q ) = 0 \operatorname{Tr}(LQ)=0 Tr ( L Q ) = 0
I 1 = a 2 + 4 b 2 + O ( ρ 4 ) , (39) I_1=a^2+4b^2+O(\rho^4), \tag*{(39)} I 1 = a 2 + 4 b 2 + O ( ρ 4 ) , (39) I 2 = a 4 + 8 a 2 b 2 + 8 b 4 + O ( ρ 5 ) , (40) I_2=a^4+8a^2b^2+8b^4+O(\rho^5), \tag*{(40)} I 2 = a 4 + 8 a 2 b 2 + 8 b 4 + O ( ρ 5 ) , (40) J F : = det ∂ ( I 1 , I 2 ) ∂ ( a , b ) = − 64 a b 3 + O ( ρ 5 ) . (41) J_{\mathrm F}:=\det\frac{\partial(I_1,I_2)}{\partial(a,b)}=-64ab^3+O(\rho^5). \tag*{(41)} J F := det ∂ ( a , b ) ∂ ( I 1 , I 2 ) = − 64 a b 3 + O ( ρ 5 ) . (41) Expanding Tr B 2 \operatorname{Tr}B^2 Tr B 2 I 1 I_1 I 1 I 2 I_2 I 2 Tr L 4 \operatorname{Tr}L^4 Tr L 4 O ( ρ 5 ) O(\rho^5) O ( ρ 5 )
I 1 , a = 2 a + O ( ρ 3 ) , I 1 , b = 8 b + O ( ρ 3 ) , I_{1,a}=2a+O(\rho^3),\qquad I_{1,b}=8b+O(\rho^3), I 1 , a = 2 a + O ( ρ 3 ) , I 1 , b = 8 b + O ( ρ 3 ) , I 2 , a = 4 a 3 + 16 a b 2 + O ( ρ 4 ) , I 2 , b = 16 a 2 b + 32 b 3 + O ( ρ 4 ) . I_{2,a}=4a^3+16ab^2+O(\rho^4),\qquad I_{2,b}=16a^2b+32b^3+O(\rho^4). I 2 , a = 4 a 3 + 16 a b 2 + O ( ρ 4 ) , I 2 , b = 16 a 2 b + 32 b 3 + O ( ρ 4 ) . Substituting these derivatives in the determinant gives (41) .
Lemma A.1 (Connected monotone level sets). Let u , v u,v u , v C 1 C^1 C 1 Q η , δ Q_{\eta,\delta} Q η , δ Theorem 4.1 . If u a , u b > 0 u_a,u_b>0 u a , u b > 0 det ∂ ( u , v ) / ∂ ( a , b ) < 0 \det\partial(u,v)/\partial(a,b)<0 det ∂ ( u , v ) / ∂ ( a , b ) < 0 ( u , v ) (u,v) ( u , v )
Proof. Suppose two points have the same value of u u u a a a b b b a 1 < a 2 a_1<a_2 a 1 < a 2 b 1 > b 2 b_1>b_2 b 1 > b 2 [ a 1 , a 2 ] × [ b 2 , b 1 ] [a_1,a_2]\times[b_2,b_1] [ a 1 , a 2 ] × [ b 2 , b 1 ] b 2 / a 2 b_2/a_2 b 2 / a 2 b 1 / a 1 b_1/a_1 b 1 / a 1 u = c u=c u = c a a a
u ( a , b 2 ) ≤ c ≤ u ( a , b 1 ) . u(a,b_2)\leq c\leq u(a,b_1). u ( a , b 2 ) ≤ c ≤ u ( a , b 1 ) . Strict monotonicity in b b b b c ( a ) b_c(a) b c ( a ) u ( a , b c ( a ) ) = c u(a,b_c(a))=c u ( a , b c ( a )) = c b c ′ = − u a / u b < 0 b_c'=-u_a/u_b<0 b c ′ = − u a / u b < 0
d d a v ( a , b c ( a ) ) = − det ∂ ( u , v ) / ∂ ( a , b ) u b > 0. (42) \frac{\mathrm{d}}{\mathrm{d}a}v(a,b_c(a))=-\frac{\det\partial(u,v)/\partial(a,b)}{u_b}>0. \tag*{(42)} d a d v ( a , b c ( a )) = − u b det ∂ ( u , v ) / ∂ ( a , b ) > 0. (42) Hence, v v v v v v
Theorem A.2 (Global uniqueness from two moments). For each fixed tail and each 0 < η < 1 0<\eta<1 0 < η < 1 ( a , b ) ↦ ( I 1 , I 2 ) (a,b)\mapsto(I_1,I_2) ( a , b ) ↦ ( I 1 , I 2 ) Q η , δ Q_{\eta,\delta} Q η , δ δ > 0 \delta>0 δ > 0
Proof. Throughout the sector, a , b ≥ c η ρ a,b \ge c_\eta\rho a , b ≥ c η ρ a b 3 ≥ κ η ρ 4 ab^3 \ge\kappa_\eta\rho^4 a b 3 ≥ κ η ρ 4
c η = η 1 + η , κ η = min η ≤ t ≤ η − 1 t 3 ( 1 + t ) 4 > 0. c_\eta= \frac{\eta}{1+\eta}, \qquad\kappa_\eta= \min_{\eta\le t \le\eta^{-1}} \frac{t^3}{(1+t)^4} > 0. c η = 1 + η η , κ η = η ≤ t ≤ η − 1 min ( 1 + t ) 4 t 3 > 0. The differentiated expansions therefore give I 1 , a , I 1 , b > 0 I_{1,a}, I_{1,b} > 0 I 1 , a , I 1 , b > 0 J F < 0 J_F < 0 J F < 0 ρ > 0 \rho> 0 ρ > 0 Lemma A.1 , applied with u = I 1 , v = I 2 u=I_1,v=I_2 u = I 1 , v = I 2
Theorem A.2 gives uniqueness from two full spectral moments, complementing the two-eigenvalue result of Theorem 4.1 . The moment pair ( I 1 , I 2 ) (I_1,I_2) ( I 1 , I 2 ) Remark 3.15 shows. For the family with F = 0 F=0 F = 0
The critical curve for the family without a tail
For F = 0 F=0 F = 0
Expansion of the moment map
The upper 2 × 2 2 \times2 2 × 2 Q Q Q (38) has only one nonzero entry, Q 22 = a 2 Q_{22}=a^2 Q 22 = a 2 u = w − 1 u=w^{-1} u = w − 1 v 0 = z − 1 v_0=z^{-1} v 0 = z − 1
1 2 ( a u v 0 + b u v 0 2 + b u 2 v 0 ) 2 , \frac{1}{2}\left(auv_0+buv_0^2+bu^2v_0\right)^2, 2 1 ( a u v 0 + b u v 0 2 + b u 2 v 0 ) 2 , whose coefficient of u 2 v 0 2 u^2v_0^2 u 2 v 0 2 a 2 / 2 a^2/2 a 2 /2 ( L 3 ) 22 = 2 a b 2 (L^3)_{22}=2ab^2 ( L 3 ) 22 = 2 a b 2
4 Tr ( L 3 Q ) = 8 a 3 b 2 . 4\operatorname{Tr}(L^3Q)=8a^3b^2. 4 Tr ( L 3 Q ) = 8 a 3 b 2 . We obtain
I 1 = a 2 + 4 b 2 + O ( ρ 4 ) , (43) I_1=a^2+4b^2+O(\rho^4), \tag*{(43)} I 1 = a 2 + 4 b 2 + O ( ρ 4 ) , (43) I 2 = a 4 + 8 a 2 b 2 + 8 b 4 + 8 a 3 b 2 + O ( ρ 6 ) , (44) I_2=a^4+8a^2b^2+8b^4+8a^3b^2+O(\rho^6), \tag*{(44)} I 2 = a 4 + 8 a 2 b 2 + 8 b 4 + 8 a 3 b 2 + O ( ρ 6 ) , (44) J 0 = − 64 a b 3 + 32 a 2 b − 192 a 3 b 2 + O ( ρ 6 ) . (45) J_0=-64ab^3+32a^2b-192a^3b^2+O(\rho^6). \tag*{(45)} J 0 = − 64 a b 3 + 32 a 2 b − 192 a 3 b 2 + O ( ρ 6 ) . (45) Differentiating 8 a 3 b 2 8a^3b^2 8 a 3 b 2 I 1 I_1 I 1 J 0 J_0 J 0
The critical curve
For F = 0 F=0 F = 0 a , b a,b a , b
I k ( a , b ) = Tr ( ( B ( a , b ) ∗ B ( a , b ) ) k ) , k = 1 , 2. I_k(a,b)=\operatorname{Tr}\left((B(a,b)^*B(a,b))^k\right), \qquad k=1,2. I k ( a , b ) = Tr ( ( B ( a , b ) ∗ B ( a , b ) ) k ) , k = 1 , 2. The proof of Lemma 3.4 shows that B B B I k I_k I k I k I_k I k a , b a,b a , b θ \theta θ
B ( e 2 i θ a , e 3 i θ b ) = D θ B ( a , b ) D θ , D θ = diag ( e i n θ ) n ≥ 1 . B(e^{2i\theta}a,e^{3i\theta}b)=D_\theta B(a,b)D_\theta,\qquad D_\theta=\operatorname{diag}(e^{in\theta})_{n\ge1}. B ( e 2 i θ a , e 3 i θ b ) = D θ B ( a , b ) D θ , D θ = diag ( e in θ ) n ≥ 1 . Unitary invariance gives I k ( a , − b ) = I k ( a , b ) I_k(a,-b)=I_k(a,b) I k ( a , − b ) = I k ( a , b ) θ = π \theta=\pi θ = π a = 0 a=0 a = 0 2 π / 3 2\pi/3 2 π /3 b b b a a a 4 π / 3 4\pi/3 4 π /3 ∂ a I k ( 0 , b ) = 0 \partial_a I_k(0,b)=0 ∂ a I k ( 0 , b ) = 0
Thus, J 0 J_0 J 0 b b b a = 0 a=0 a = 0 q ( a , y ) q(a,y) q ( a , y ) ( 0 , 0 ) (0,0) ( 0 , 0 )
J 0 ( a , b ) = a b q ( a , b 2 ) . (46) J_0(a,b)=abq(a,b^2). \tag*{(46)} J 0 ( a , b ) = ab q ( a , b 2 ) . (46) Equation (45) gives
q ( a , y ) = 32 a 3 − 64 y − 192 a y + O ( a 4 + a 2 ∣ y ∣ + y 2 ) . (47) q(a,y)=32a^3-64y-192ay+O(a^4+a^2|y|+y^2). \tag*{(47)} q ( a , y ) = 32 a 3 − 64 y − 192 a y + O ( a 4 + a 2 ∣ y ∣ + y 2 ) . (47) Since q y ( 0 , 0 ) = − 64 q_y(0,0)=-64 q y ( 0 , 0 ) = − 64 h h h
q ( a , h ( a ) ) = 0 , h ( a ) = 1 2 a 3 + O ( a 4 ) . (48) q(a,h(a))=0,\qquad h(a)=\frac{1}{2}a^3+O(a^4). \tag*{(48)} q ( a , h ( a )) = 0 , h ( a ) = 2 1 a 3 + O ( a 4 ) . (48) For sufficiently small positive a a a h ( a ) > 0 h(a)>0 h ( a ) > 0 h ′ ( a ) = 3 2 a 2 + O ( a 3 ) > 0 h'(a)=\frac{3}{2}a^2+O(a^3)>0 h ′ ( a ) = 2 3 a 2 + O ( a 3 ) > 0 q y < 0 q_y<0 q y < 0
q ( a , y ) = u ( a , y ) ( h ( a ) − y ) , u ( a , y ) > 0 , q(a,y)=u(a,y)(h(a)-y),\qquad u(a,y)>0, q ( a , y ) = u ( a , y ) ( h ( a ) − y ) , u ( a , y ) > 0 , gives
sgn J 0 ( a , b ) = sgn ( h ( a ) − b 2 ) ( a , b > 0 small ) . (49) \operatorname{sgn}J_0(a,b)=\operatorname{sgn}(h(a)-b^2)\qquad(a,b>0\text{ small}). \tag*{(49)} sgn J 0 ( a , b ) = sgn ( h ( a ) − b 2 ) ( a , b > 0 small ) . (49) Hence, b 2 = h ( a ) b^2=h(a) b 2 = h ( a )
Theorem A.3 (Global uniqueness on either side of the fold). There is δ 0 > 0 \delta_0>0 δ 0 > 0
U − = { 0 < a , b < δ 0 : b 2 > h ( a ) } , U + = { 0 < a , b < δ 0 : b 2 < h ( a ) } . U_-=\{0<a,b<\delta_0:b^2>h(a)\},\qquad U_+=\{0<a,b<\delta_0:b^2<h(a)\}. U − = { 0 < a , b < δ 0 : b 2 > h ( a )} , U + = { 0 < a , b < δ 0 : b 2 < h ( a )} . Thus, the NP spectrum determines the normalised domain with F = 0 F=0 F = 0 δ 0 \delta_0 δ 0
0 < a , b < δ 0 , b 2 ≥ a 3 , and 0 < a , b < δ 0 , b 2 ≤ 1 4 a 3 . 0<a,b<\delta_0,\quad b^2\ge a^3,\qquad\text{and}\qquad0<a,b<\delta_0,\quad b^2\le\frac{1}{4}a^3. 0 < a , b < δ 0 , b 2 ≥ a 3 , and 0 < a , b < δ 0 , b 2 ≤ 4 1 a 3 . Proof. For real a , b > 0 a,b>0 a , b > 0 a , b a,b a , b B 11 = a B_{11}=a B 11 = a B 12 = B 21 = 2 b B_{12}=B_{21}=\sqrt{2}b B 12 = B 21 = 2 b
I 1 , a ≥ 2 a > 0 , I 1 , b ≥ 8 b > 0. I_{1,a}\ge2a>0,\qquad I_{1,b}\ge8b>0. I 1 , a ≥ 2 a > 0 , I 1 , b ≥ 8 b > 0. Distinct points on the same I 1 I_1 I 1 b c ( a ) b_c(a) b c ( a ) h h h b c ( a ) 2 − h ( a ) b_c(a)^2-h(a) b c ( a ) 2 − h ( a ) (42) and (49) then imply strict monotonicity of I 2 I_2 I 2 h ( a ) = a 3 / 2 + O ( a 4 ) h(a)=a^3/2+O(a^4) h ( a ) = a 3 /2 + O ( a 4 ) a a a
Proposition A.4 (The two moments do not distinguish the whole quadrant). For every sufficiently small a 0 > 0 a_0>0 a 0 > 0 ( a 0 , b 0 ) (a_0,b_0) ( a 0 , b 0 ) b 0 = h ( a 0 ) > 0 b_0=\sqrt{h(a_0)}>0 b 0 = h ( a 0 ) > 0 b 2 = h ( a ) b^2=h(a) b 2 = h ( a ) I 1 I_1 I 1 I 2 I_2 I 2
Proof. The level set of I 1 I_1 I 1 ( a 0 , b 0 ) (a_0,b_0) ( a 0 , b 0 ) b = b c ( a ) b=b_c(a) b = b c ( a ) b c ′ = − I 1 , a / I 1 , b < 0 b'_c=-I_{1,a}/I_{1,b}<0 b c ′ = − I 1 , a / I 1 , b < 0 J 0 ( a , b ) = a b u ( a , b 2 ) ( h ( a ) − b 2 ) J_0(a,b)=abu(a,b^2)(h(a)-b^2) J 0 ( a , b ) = ab u ( a , b 2 ) ( h ( a ) − b 2 )
d d a J 0 ( a , b c ( a ) ) ∣ a = a 0 = a 0 b 0 u ( a 0 , b 0 2 ) ( h ′ ( a 0 ) − 2 b 0 b c ′ ( a 0 ) ) > 0. \left.\frac{\mathrm{d}}{\mathrm{d}a}J_0(a,b_c(a))\right|_{a=a_0}
=a_0b_0u(a_0,b_0^2)\bigl(h'(a_0)-2b_0b'_c(a_0)\bigr)>0. d a d J 0 ( a , b c ( a )) a = a 0 = a 0 b 0 u ( a 0 , b 0 2 ) ( h ′ ( a 0 ) − 2 b 0 b c ′ ( a 0 ) ) > 0. By Equation (42) , the second derivative of I 2 ( a , b c ( a ) ) I_2(a,b_c(a)) I 2 ( a , b c ( a )) a 0 a_0 a 0
These pairs have equal values of two moments; higher moments may distinguish them. Hence the two-moment map is not injective on U − ∪ U + U_-\cup U_+ U − ∪ U +
Uniqueness from polarization data
Spectral weights and polarization data
Complete weighted NP data recover all area moments and hence the domain. At one contrast, all uncontracted GPTs also give uniqueness for analytic Jordan domains. This second proof uses all polynomial observations; [5 Theorem 3.1] requires only harmonic combinations. The spectral weights record the excitation and observation couplings of energy-normalised eigenfunctions.
Let D D D A D A_D A D H H H G G G
g H = ∂ ν H , ℓ G ( φ ) = ∫ ∂ D G φ d s . g_H=\partial_\nu H,\qquad\ell_G(\varphi)=\int_{\partial D}G\varphi\,\mathrm{d}s. g H = ∂ ν H , ℓ G ( φ ) = ∫ ∂ D Gφ d s . The input has zero mean. Equivalence of the energy norm with the mean-zero H − 1 / 2 H^{-1/2} H − 1/2 ℓ G \ell_G ℓ G G G G
Let E μ E_\mu E μ A D A_D A D μ \mu μ E 0 E_0 E 0
W G , H ( μ ; D ) = ℓ G ( E μ g H ) . (50) W_{G,H}(\mu;D)=\ell_G(E_\mu g_H). \tag*{(50)} W G , H ( μ ; D ) = ℓ G ( E μ g H ) . (50) If ( φ μ , j ) j (\varphi_{\mu,j})_j ( φ μ , j ) j
W G , H ( μ ; D ) = ∑ j ( g H , φ μ , j ) ∗ ℓ G ( φ μ , j ) . W_{G,H}(\mu;D)=\sum_j (g_H,\varphi_{\mu,j})_*\,\ell_G(\varphi_{\mu,j}). W G , H ( μ ; D ) = j ∑ ( g H , φ μ , j ) ∗ ℓ G ( φ μ , j ) . These weights are independent of the chosen basis and are determined by the eigenvalues, energy-normalised eigenfunctions and their geometric couplings.
A direct area-moment proof from weighted spectral data
Theorem B.1 (Uniqueness from weighted spectral data). The reduced NP eigenvalues and the weights (50), including the weights of the zero eigenspace, for all harmonic polynomials H , G H,G H , G
Proof. The spectral theorem and the continuity of ℓ G \ell_G ℓ G
M H , G ( λ ; D ) = ℓ G ( ( λ I − A D ) − 1 g H ) = ∑ μ W G , H ( μ ; D ) λ − μ . (51) M_{H,G}(\lambda;D)=\ell_G\left((\lambda I-A_D)^{-1}g_H\right)=\sum_\mu\frac{W_{G,H}(\mu;D)}{\lambda-\mu}. \tag*{(51)} M H , G ( λ ; D ) = ℓ G ( ( λ I − A D ) − 1 g H ) = μ ∑ λ − μ W G , H ( μ ; D ) . (51) Absolute summability of the weights follows by representing ℓ G \ell_G ℓ G g H g_H g H
λ ( λ I − A D ) − 1 = ( I − A D / λ ) − 1 ⟶ I ( λ → + ∞ ) . \lambda(\lambda I-A_D)^{-1}=(I-A_D/\lambda)^{-1}\longrightarrow I\qquad(\lambda\to+\infty). λ ( λ I − A D ) − 1 = ( I − A D / λ ) − 1 ⟶ I ( λ → + ∞ ) . Green’s formula then gives
lim λ → + ∞ λ M H , G ( λ ; D ) = ∫ ∂ D G ∂ ν H d s = ∫ D ∇ G ⋅ ∇ H d A , (52) \lim_{\lambda\to+\infty}\lambda M_{H,G}(\lambda;D)=\int_{\partial D}G\partial_\nu H\,\mathrm{d}s=\int_D\nabla G\cdot\nabla H\,\mathrm{d}A, \tag*{(52)} λ → + ∞ lim λ M H , G ( λ ; D ) = ∫ ∂ D G ∂ ν H d s = ∫ D ∇ G ⋅ ∇ H d A , (52) since Δ H = 0 \Delta H=0 Δ H = 0
For integers p , q ≥ 0 p,q\ge0 p , q ≥ 0
H p ( z ) = z p + 1 p + 1 , G q ( z ) = z ˉ q + 1 q + 1 . H_p(z)=\frac{z^{p+1}}{p+1},\qquad G_q(z)=\frac{\bar{z}^{q+1}}{q+1}. H p ( z ) = p + 1 z p + 1 , G q ( z ) = q + 1 z ˉ q + 1 . These are complex harmonic polynomials satisfying ∇ H p ⋅ ∇ G q = 2 z p z ˉ q \nabla H_p\cdot\nabla G_q=2z^p\bar{z}^q ∇ H p ⋅ ∇ G q = 2 z p z ˉ q
∫ D z p z ˉ q d A = 1 2 lim λ → + ∞ λ M H p , G q ( λ ; D ) = 1 2 ∑ μ W G q , H p ( μ ; D ) . (53) \int_D z^p\bar{z}^q\,\mathrm{d}A=\frac{1}{2}\lim_{\lambda\to+\infty}\lambda M_{H_p,G_q}(\lambda;D)=\frac{1}{2}\sum_\mu W_{G_q,H_p}(\mu;D). \tag*{(53)} ∫ D z p z ˉ q d A = 2 1 λ → + ∞ lim λ M H p , G q ( λ ; D ) = 2 1 μ ∑ W G q , H p ( μ ; D ) . (53) By bilinear complexification, real harmonic data suffice to recover these mixed moments. If two domains have the same weighted spectral data, their area measures agree on all polynomials in z , z ˉ z,\bar{z} z , z ˉ x , y x,y x , y D = int D ‾ D=\operatorname{int}\overline{D} D = int D
Remark B.2. By Formula (53), the sums of the complete weights suffice to determine the domain. The eigenvalues affect the full resolvent response; its leading coefficient at infinity depends only on these sums. The kernel contribution is essential: the reduced NP operator of the disk is zero, but its weights recover nonzero area moments. Here the complete spectral decomposition allows λ \lambda λ
A direct proof for GPTs at one fixed contrast
At a fixed contrast, the moments of all polynomial observations determine the boundary charge measures. This yields uniqueness in the analytic Jordan class once we establish density of the harmonic inputs.
Lemma B.3 (Density of harmonic-polynomial Neumann traces). For a bounded simply connected planar domain with analytic Jordan boundary,
span { ∂ ν H : H a harmonic polynomial } \operatorname{span}\{\partial_\nu H:H\text{ a harmonic polynomial}\} span { ∂ ν H : H a harmonic polynomial } is dense in L 0 2 ( ∂ D ) L^2_0(\partial D) L 0 2 ( ∂ D )
Proof. It suffices to approximate a smooth real mean-zero function f f f L 0 2 L^2_0 L 0 2 u ∈ C ∞ ( D ‾ ) u\in C^\infty(\overline{D}) u ∈ C ∞ ( D ) ∂ ν u = f \partial_\nu u=f ∂ ν u = f h h h u u u ψ : D → D \psi:\mathbb{D}\to D ψ : D → D ϕ = ψ − 1 \phi=\psi^{-1} ϕ = ψ − 1 0 < r < 1 0<r<1 0 < r < 1
h r ( z ) = h ( ψ ( r ϕ ( z ) ) ) h_r(z)=h(\psi(r\phi(z))) h r ( z ) = h ( ψ ( r ϕ ( z ))) is holomorphic in a neighbourhood of D ‾ \overline{D} D h h h C 1 ( D ‾ ) C^1(\overline{D}) C 1 ( D ) r ↑ 1 r\uparrow1 r ↑ 1 D ‾ \overline{D} D h r h_r h r D ‾ \overline{D} D u u u C 1 ( D ‾ ) C^1(\overline{D}) C 1 ( D ) L 2 L^2 L 2 f f f
Theorem B.4 (Fixed-contrast uniqueness from uncontracted GPTs). Let D 1 , D 2 D_1,D_2 D 1 , D 2 λ ( 1 ) , λ ( 2 ) ∈ R ∖ [ − 1 / 2 , 1 / 2 ] \lambda^{(1)},\lambda^{(2)} \in\mathbb{R} \setminus[-1/2,1/2] λ ( 1 ) , λ ( 2 ) ∈ R ∖ [ − 1/2 , 1/2 ]
∫ ∂ D 1 G ( λ ( 1 ) I − K D 1 ∗ ) − 1 [ ∂ ν H ] d s = ∫ ∂ D 2 G ( λ ( 2 ) I − K D 2 ∗ ) − 1 [ ∂ ν H ] d s \int_{\partial D_1} G(\lambda^{(1)}I-K_{D_1}^*)^{-1}[\partial_\nu H]\,\mathrm{d}s = \int_{\partial D_2} G(\lambda^{(2)}I-K_{D_2}^*)^{-1}[\partial_\nu H]\,\mathrm{d}s ∫ ∂ D 1 G ( λ ( 1 ) I − K D 1 ∗ ) − 1 [ ∂ ν H ] d s = ∫ ∂ D 2 G ( λ ( 2 ) I − K D 2 ∗ ) − 1 [ ∂ ν H ] d s for every harmonic polynomial H H H G G G D 1 = D 2 D_1=D_2 D 1 = D 2 λ ( 1 ) = λ ( 2 ) \lambda^{(1)}=\lambda^{(2)} λ ( 1 ) = λ ( 2 )
Proof. Linear combinations of input and observation monomials show that equality of all uncontracted GPTs at the respective contrasts gives the stated hypothesis. For each H H H
φ j , H = ( λ ( j ) I − K D j ∗ ) − 1 [ ∂ ν H ] , μ j , H = φ j , H d s ∂ D j . \varphi_{j,H}=(\lambda^{(j)}I-K_{D_j}^*)^{-1}[\partial_\nu H],\qquad\mu_{j,H}=\varphi_{j,H}\,\mathrm{d}s_{\partial D_j}. φ j , H = ( λ ( j ) I − K D j ∗ ) − 1 [ ∂ ν H ] , μ j , H = φ j , H d s ∂ D j . The resolvents are bounded isomorphisms on L 2 ( ∂ D j ) L^2(\partial D_j) L 2 ( ∂ D j ) L 2 L^2 L 2 μ 1 , H , μ 2 , H \mu_{1,H},\mu_{2,H} μ 1 , H , μ 2 , H
μ 1 , H = μ 2 , H \mu_{1,H}=\mu_{2,H} μ 1 , H = μ 2 , H as measures on the ambient plane, for every H H H
By Lemma B.3 and invertibility of the resolvents, the responses φ j , H \varphi_{j,H} φ j , H L 2 ( ∂ D j ) L^2(\partial D_j) L 2 ( ∂ D j ) x ∈ ∂ D 1 ∖ ∂ D 2 x\in\partial D_1\setminus\partial D_2 x ∈ ∂ D 1 ∖ ∂ D 2 x x x ∂ D 2 \partial D_2 ∂ D 2 ∂ D 1 \partial D_1 ∂ D 1 φ 1 , H \varphi_{1,H} φ 1 , H ∂ D 1 ⊂ ∂ D 2 \partial D_1\subset\partial D_2 ∂ D 1 ⊂ ∂ D 2
On the common boundary, equality of the charge measures gives φ 1 , H = φ 2 , H = φ H \varphi_{1,H}=\varphi_{2,H}=\varphi_H φ 1 , H = φ 2 , H = φ H ( λ ( 1 ) − λ ( 2 ) ) φ H = 0 (\lambda^{(1)}-\lambda^{(2)})\varphi_H=0 ( λ ( 1 ) − λ ( 2 ) ) φ H = 0 H = x 1 H=x_1 H = x 1 φ H ≠ 0 \varphi_H\ne0 φ H = 0 λ ( 1 ) = λ ( 2 ) \lambda^{(1)}=\lambda^{(2)} λ ( 1 ) = λ ( 2 ) λ = λ ( 1 ) = λ ( 2 ) \lambda=\lambda^{(1)}=\lambda^{(2)} λ = λ ( 1 ) = λ ( 2 ) k = ( 2 λ + 1 ) / ( 2 λ − 1 ) k=(2\lambda+1)/(2\lambda-1) k = ( 2 λ + 1 ) / ( 2 λ − 1 )
Additional uniqueness classes and extensions
The class in Theorem 3.2 admits several extensions: other finite-dimensional slices with injective spectral differential, symmetry-reduced families, and saturation under spectrum-preserving transformations. Saturation gives uniqueness on a quotient. Throughout this appendix, eigenvalues are ordered with multiplicity, as in Subsection 3.7.
Ellipses: an exact global one-parameter class
Proposition C.1 (Normalised ellipses). Let E t E_t E t Ψ t ( w ) = w + t / w \Psi_t(w)=w+t/w Ψ t ( w ) = w + t / w 0 ≤ t < 1 0\le t<1 0 ≤ t < 1 1 + t 1+t 1 + t 1 − t 1-t 1 − t
λ n ( E t ) = 1 2 t n , n ≥ 1. \lambda_n(E_t)=\frac{1}{2}t^n,\qquad n\ge1. λ n ( E t ) = 2 1 t n , n ≥ 1. At t = 0 t=0 t = 0
∣ t − u ∣ = 2 ∣ λ 1 ( E t ) − λ 1 ( E u ) ∣ , d H ( E t ‾ , E u ‾ ) = ∣ t − u ∣ . |t-u|=2|\lambda_1(E_t)-\lambda_1(E_u)|,\qquad d_H(\overline{E_t},\overline{E_u})=|t-u|. ∣ t − u ∣ = 2∣ λ 1 ( E t ) − λ 1 ( E u ) ∣ , d H ( E t , E u ) = ∣ t − u ∣. The NP spectrum therefore determines an ellipse up to similarities, with global Lipschitz stability of its capacity-normalised semiaxes. The axis ratio ( 1 + t ) / ( 1 − t ) (1+t)/(1-t) ( 1 + t ) / ( 1 − t ) λ 1 \lambda_1 λ 1 t ≤ t ∗ < 1 t\le t_*<1 t ≤ t ∗ < 1
Proof. The logarithmic kernel is
− log Ψ t ( w ) − Ψ t ( ζ ) w − ζ = − log ( 1 − t / ( w ζ ) ) = ∑ n ≥ 1 t n n w − n ζ − n . -\log\frac{\Psi_t(w)-\Psi_t(\zeta)}{w-\zeta}=-\log(1-t/(w\zeta))=\sum_{n\geq1}\frac{t^n}{n}w^{-n}\zeta^{-n}. − log w − ζ Ψ t ( w ) − Ψ t ( ζ ) = − log ( 1 − t / ( w ζ )) = n ≥ 1 ∑ n t n w − n ζ − n . The normalised Grunsky matrix is therefore diag ( t , t 2 , … ) \operatorname{diag}(t,t^2,\ldots) diag ( t , t 2 , … ) (25) gives the eigenvalues. Corresponding boundary points differ by ( t − u ) e − i θ (t-u)e^{-i\theta} ( t − u ) e − i θ ∣ t − u ∣ |t-u| ∣ t − u ∣ ∣ t − u ∣ |t-u| ∣ t − u ∣ 2 / ( 1 − t ) 2 2/(1-t)^2 2/ ( 1 − t ) 2
Single-mode families, including repeated leading eigenvalues
For fixed k k k t ≥ 0 t\geq0 t ≥ 0 [4 ] , equation (2.9), has capacity-normalised map Ψ t ( w ) = w + t w − k \Psi_t(w)=w+tw^{-k} Ψ t ( w ) = w + t w − k k k k m m m
Theorem C.2 (Exact single-mode recovery). Fix k ≥ 1 k\geq1 k ≥ 1
Ψ t ( w ) = w + t w − k + R ( t ; w ) , R ( t ; w ) = ∑ n > k r n ( t ) w − n , \Psi_t(w)=w+tw^{-k}+\mathcal{R}(t;w),\qquad\mathcal{R}(t;w)=\sum_{n>k}r_n(t)w^{-n}, Ψ t ( w ) = w + t w − k + R ( t ; w ) , R ( t ; w ) = n > k ∑ r n ( t ) w − n , where R \mathcal{R} R t = 0 t=0 t = 0 ∣ w ∣ > r ∗ |w|>r_\ast ∣ w ∣ > r ∗ 0 < r ∗ < 1 0<r_\ast<1 0 < r ∗ < 1 t t t O ( t 2 ) O(t^2) O ( t 2 ) r ∗ < ∣ w ∣ = r < 1 r_\ast<|w|=r<1 r ∗ < ∣ w ∣ = r < 1 δ , c , C > 0 \delta,c,C>0 δ , c , C > 0 0 ≤ t ≤ δ 0\leq t\leq\delta 0 ≤ t ≤ δ t , u ∈ [ 0 , δ ] t,u\in[0,\delta] t , u ∈ [ 0 , δ ]
c ∣ t − u ∣ ≤ ( ∑ j = 1 k ∣ λ j ( D t ) − λ j ( D u ) ∣ 2 ) 1 / 2 ≤ C ∣ t − u ∣ . (54) c|t-u|\leq\left(\sum_{j=1}^{k}\left|\lambda_j(D_t)-\lambda_j(D_u)\right|^2\right)^{1/2}\leq C|t-u|. \tag*{(54)} c ∣ t − u ∣ ≤ ( j = 1 ∑ k ∣ λ j ( D t ) − λ j ( D u ) ∣ 2 ) 1/2 ≤ C ∣ t − u ∣. (54) The single scalar observable
J k ( t ) = ( ∑ j = 1 k λ j ( D t ) 2 ) 1 / 2 J_k(t)=\left(\sum_{j=1}^{k}\lambda_j(D_t)^2\right)^{1/2} J k ( t ) = ( j = 1 ∑ k λ j ( D t ) 2 ) 1/2 is strictly increasing and has a Lipschitz inverse, uniformly down to the disk. For t , u ∈ [ 0 , δ ] t,u\in[0,\delta] t , u ∈ [ 0 , δ ] ℓ s 2 \ell_s^2 ℓ s 2 s ≥ 0 s\geq0 s ≥ 0 ∣ t − u ∣ |t-u| ∣ t − u ∣ s s s
Proof. The estimates in Subsections 3.2 and 3.3 apply to this family. In the real self-adjoint Grunsky realisation,
B ( t ) = t A k + O H S ( t 2 ) , ( A k ) i j = { i j , i + j = k + 1 , 0 , otherwise . B(t)=tA_k+O_{\mathrm{HS}}(t^2),\qquad(A_k)_{ij}=
\begin{cases}
\sqrt{ij}, & i+j=k+1,\\
0, & \text{otherwise}.
\end{cases} B ( t ) = t A k + O HS ( t 2 ) , ( A k ) ij = { ij , 0 , i + j = k + 1 , otherwise . The remainder has derivative O H S ( t ) O_{\mathrm{HS}}(t) O HS ( t ) A k A_k A k k k k k k k
χ k = ∥ A k ∥ H S 2 = ∑ j = 1 k j ( k + 1 − j ) = k ( k + 1 ) ( k + 2 ) 6 . \chi_k=\|A_k\|_{\mathrm{HS}}^2=\sum_{j=1}^{k}j(k+1-j)=\frac{k(k+1)(k+2)}{6}. χ k = ∥ A k ∥ HS 2 = j = 1 ∑ k j ( k + 1 − j ) = 6 k ( k + 1 ) ( k + 2 ) . Define C ( t ) = B ( t ) / t C(t)=B(t)/t C ( t ) = B ( t ) / t C ( 0 ) = A k C(0)=A_k C ( 0 ) = A k A k A_k A k P ( t ) P(t) P ( t ) k k k t t t k k k C ( t ) C(t) C ( t )
f ( t ) = Tr ( C ( t ) 2 P ( t ) ) = χ k + O ( t ) f(t)=\operatorname{Tr}\left(C(t)^2P(t)\right)=\chi_k+O(t) f ( t ) = Tr ( C ( t ) 2 P ( t ) ) = χ k + O ( t ) is analytic, real and positive near zero, with f ′ ( t ) = O ( 1 ) f'(t)=O(1) f ′ ( t ) = O ( 1 ) t ≥ 0 t\ge0 t ≥ 0
J k ( t ) = t 2 f ( t ) , J k ′ ( t ) = 1 2 χ k + O ( t ) . \mathcal{J}_k(t)=\frac{t}{2}\sqrt{f(t)},\qquad\mathcal{J}_k'(t)=\frac{1}{2}\sqrt{\chi_k}+O(t). J k ( t ) = 2 t f ( t ) , J k ′ ( t ) = 2 1 χ k + O ( t ) . Reducing δ \delta δ (54) . The upper bound follows from the Hilbert–Schmidt singular-value inequality and ∥ B ( t ) − B ( u ) ∥ H S ≤ C ∣ t − u ∣ \lVert B(t)-B(u)\rVert_{\mathrm{HS}}\le C\lvert t-u\rvert ∥ B ( t ) − B ( u ) ∥ HS ≤ C ∣ t − u ∣
Corresponding boundary points give the geometric upper bound. For the lower bound, use the support-function expansion
h t ( θ ) = 1 + t cos ( ( k + 1 ) θ ) + q t ( θ ) , ∥ q t − q u ∥ C 0 ≤ C δ ∣ t − u ∣ . h_t(\theta)=1+t\cos((k+1)\theta)+q_t(\theta),\qquad\lVert q_t-q_u\rVert_{C^0}\le C\delta\lvert t-u\rvert. h t ( θ ) = 1 + t cos (( k + 1 ) θ ) + q t ( θ ) , ∥ q t − q u ∥ C 0 ≤ C δ ∣ t − u ∣ . This follows from the calculation in Proposition 3.3 : the linear normal displacement is the displayed cosine, and the remainder has derivative O ( t ) O(t) O ( t ) δ \delta δ O ( t ) O(t) O ( t ) w − k w^{-k} w − k t t t
For k ≥ 2 k\ge2 k ≥ 2
t 2 j ( k + 1 − j ) , j = 1 , … , k , \frac{t}{2}\sqrt{j(k+1-j)},\qquad j=1,\ldots,k, 2 t j ( k + 1 − j ) , j = 1 , … , k , after rearrangement in nonincreasing order. This is the first-order spectral pattern of [4 Theorems 2.2–2.3] . The cluster argument handles the repetitions and gives a uniform inverse estimate for exact eigenvalues. The mode k k k t = 0 t=0 t = 0 k k k
Comparison with plasmonic particle shape recovery
Ammari, Putinar, Ruiz, Yu, and Zhang [4 ] study a single-particle family with normalised boundary e i θ + t e − i k θ e^{i\theta}+te^{-ik\theta} e i θ + t e − ik θ m m m m m m
Their Theorems 2.2–2.3 give first-order NP eigenvalue expansions and leading-order eigenfunctions. For k ≥ 3 k\ge3 k ≥ 3
ℓ = t 2 k , ℓ ′ = t 2 2 ( k − 1 ) . \ell=\frac{t}{2}\sqrt{k},\qquad\ell'=\frac{t}{2}\sqrt{2(k-1)}. ℓ = 2 t k , ℓ ′ = 2 t 2 ( k − 1 ) . These give the algebraic inversion
k = ℓ 2 ℓ 2 − ( ℓ ′ ) 2 / 2 , t = 2 ℓ 2 − ( ℓ ′ ) 2 / 2 . k=\frac{\ell^2}{\ell^2-(\ell')^2/2},\qquad t=2\sqrt{\ell^2-(\ell')^2/2}. k = ℓ 2 − ( ℓ ′ ) 2 /2 ℓ 2 , t = 2 ℓ 2 − ( ℓ ′ ) 2 /2 . For k = 1 k=1 k = 1 ℓ ′ = 0 \ell'=0 ℓ ′ = 0 k = 2 k=2 k = 2 k ≥ 3 k\ge3 k ≥ 3
Our injectivity and uniform Lipschitz estimates on specified small families rely on controlling the nonlinear remainder and counting eigenvalues with multiplicity. The response analysis in [4 ] identifies the modes that couple to low-order polarization measurements and recovers size from amplitudes in the leading model, adding information beyond eigenvalues. Its Theorem 3.2 gives an exact spectrum for two nearly touching disks. For equal radius r > 0 r>0 r > 0 g > 0 g>0 g > 0
λ + , 2 n − 1 = λ + , 2 n = 1 2 e − 2 n ξ 0 , n ≥ 1 , cosh ξ 0 = 1 + g 2 r . \lambda_{+,2n-1}=\lambda_{+,2n}=\frac{1}{2}e^{-2n\xi_0},\qquad n\ge1,\qquad\cosh\xi_0=1+\frac{g}{2r}. λ + , 2 n − 1 = λ + , 2 n = 2 1 e − 2 n ξ 0 , n ≥ 1 , cosh ξ 0 = 1 + 2 r g . Each positive and negative level has multiplicity two, corresponding to the cosine and sine bipolar modes. Hence g / r = 2 { cosh [ − 1 2 log ( 2 λ + , 1 ) ] − 1 } g/r=2\{\cosh[-\frac{1}{2}\log(2\lambda_{+,1})]-1\} g / r = 2 { cosh [ − 2 1 log ( 2 λ + , 1 )] − 1 }
A local criterion and prescribed perturbations
Theorem C.3 (Finite-dimensional local spectral criterion). Let p ↦ D p p\mapsto D_p p ↦ D p p ∈ U p\in U p ∈ U normalised family on an open set U ⊂ R d U\subset\mathbb{R}^d U ⊂ R d F : U → R N \mathcal{F}:U\to\mathbb{R}^N F : U → R N C 1 C^1 C 1 N ≥ d N\ge d N ≥ d p 0 ∈ U p_0\in U p 0 ∈ U D F ( p 0 ) D\mathcal{F}(p_0) D F ( p 0 ) d d d V V V p 0 p_0 p 0
c ∣ p − q ∣ ≤ ∣ F ( p ) − F ( q ) ∣ ≤ C ∣ p − q ∣ . c|p-q|\leq|\mathcal{F}(p)-\mathcal{F}(q)|\leq C|p-q|. c ∣ p − q ∣ ≤ ∣ F ( p ) − F ( q ) ∣ ≤ C ∣ p − q ∣. If the boundary parametrisations are locally Lipschitz in p p p C 0 C^0 C 0 d H ( ∂ D p , ∂ D q ) ≤ C ′ ∣ F ( p ) − F ( q ) ∣ d_H(\partial D_p,\partial D_q)\leq C'|\mathcal{F}(p)-\mathcal{F}(q)| d H ( ∂ D p , ∂ D q ) ≤ C ′ ∣ F ( p ) − F ( q ) ∣
Proof. Let A = D F ( p 0 ) A=D\mathcal{F}(p_0) A = D F ( p 0 ) σ > 0 \sigma>0 σ > 0 ∥ D F ( p ) − A ∥ o p ≤ σ / 2 \|D\mathcal{F}(p)-A\|_{\mathrm{op}}\leq\sigma/2 ∥ D F ( p ) − A ∥ op ≤ σ /2 q q q p p p
F ( p ) − F ( q ) = A ( p − q ) + e , ∣ e ∣ ≤ σ 2 ∣ p − q ∣ . \mathcal{F}(p)-\mathcal{F}(q)=A(p-q)+e,\qquad|e|\leq\frac{\sigma}{2}|p-q|. F ( p ) − F ( q ) = A ( p − q ) + e , ∣ e ∣ ≤ 2 σ ∣ p − q ∣. The smallest singular value of A A A D F D\mathcal{F} D F
One may use isolated simple eigenvalue branches or sums of powers over isolated clusters containing multiple eigenvalues. The rank condition must be checked in each case. With normalisation fixed, the criterion also covers nonsymmetric conformal families and families with complex coefficients, provided the spectral differential has full real rank.
The criterion persists under prescribed perturbations. If ∥ D F − A ∥ o p ≤ σ / 4 \|D\mathcal{F}-A\|_{\mathrm{op}}\leq\sigma/4 ∥ D F − A ∥ op ≤ σ /4 V V V C 1 C^1 C 1 F ~ \widetilde{\mathcal{F}} F ∥ D F ~ − D F ∥ o p ≤ σ / 4 \|D\widetilde{\mathcal{F}}-D\mathcal{F}\|_{\mathrm{op}}\leq\sigma/4 ∥ D F − D F ∥ op ≤ σ /4 σ / 2 \sigma/2 σ /2 d d d d d d d d d
Möbius saturation and quotient stability
Möbius transformations generally change the normalisation, the prescribed Laurent coefficient relations, and convexity. They still give an invariant enlargement of a normalised uniqueness class.
For a smooth Jordan domain D D D T T T ∂ D \partial D ∂ D T # D T_{\#}D T # D T ( ∂ D ) T(\partial D) T ( ∂ D ) A = { D a : a ∈ V } \mathcal{A} = \{D_a : a \in V\} A = { D a : a ∈ V } V ⊂ R d V \subset\mathbb{R}^{d} V ⊂ R d
Sat ( A ) = { T # D a : a ∈ V , T extended M o ¨ bius, pole off ∂ D a } . \operatorname{Sat}(\mathcal{A}) = \{T_{\#}D_a : a \in V,\ T\text{ extended Möbius, pole off }\partial D_a\}. Sat ( A ) = { T # D a : a ∈ V , T extended M o ¨ bius, pole off ∂ D a } . Lemma C.4 (Reduced planar spectral invariance). For every smooth Jordan domain D D D T T T D D D T # D T_{\#}D T # D
Proof. For T ( z ) = ( a z + b ) / ( c z + d ) T(z) = (az + b)/(cz + d) T ( z ) = ( a z + b ) / ( cz + d )
log ∣ T ( x ) − T ( y ) ∣ = log ∣ x − y ∣ + log ∣ a d − b c ∣ − log ∣ c x + d ∣ − log ∣ c y + d ∣ . \log|T(x) - T(y)| = \log|x-y| + \log|ad-bc| - \log|cx+d| - \log|cy+d|. log ∣ T ( x ) − T ( y ) ∣ = log ∣ x − y ∣ + log ∣ a d − b c ∣ − log ∣ c x + d ∣ − log ∣ cy + d ∣. Define the charge-preserving boundary transport by ( U φ ) ( T ( x ) ) = φ ( x ) / ∣ T ′ ( x ) ∣ (U\varphi)(T(x)) = \varphi(x)/|T'(x)| ( U φ ) ( T ( x )) = φ ( x ) /∣ T ′ ( x ) ∣ x x x y y y x x x
K T # D ∗ U = ϵ U K D ∗ , K^{*}_{T_{\#}D}U = \epsilon U K^{*}_{D}, K T # D ∗ U = ϵ U K D ∗ , where ϵ = 1 \epsilon= 1 ϵ = 1 T ( D ) T(D) T ( D ) ϵ = − 1 \epsilon= -1 ϵ = − 1 U U U μ ↔ − μ \mu\leftrightarrow-\mu μ ↔ − μ
Theorem C.5 (Uniqueness and stability modulo Möbius transformations). Suppose A \mathcal{A} A Λ ( D a ) \Lambda(D_a) Λ ( D a )
∣ a − b ∣ ≤ C ∣ Λ ( D a ) − Λ ( D b ) ∣ . |a-b| \le C|\Lambda(D_a)-\Lambda(D_b)|. ∣ a − b ∣ ≤ C ∣Λ ( D a ) − Λ ( D b ) ∣. Then the NP spectrum determines each domain in Sat ( A ) \operatorname{Sat}(\mathcal{A}) Sat ( A ) A \mathcal{A} A
d A ( [ T # D a ] , [ S # D b ] ) = ∣ a − b ∣ . d_{\mathcal{A}}([T_{\#}D_a],[S_{\#}D_b]) = |a-b|. d A ([ T # D a ] , [ S # D b ]) = ∣ a − b ∣. This metric is well defined and satisfies the same inverse Lipschitz bound in the spectral data. If the original class also has the forward spectral estimate, the quotient parameter metric and spectral distance are bi-Lipschitz equivalent.
Proof. Equal spectra for T # D a T_{\#}D_a T # D a S # D b S_{\#}D_b S # D b D a , D b D_a,D_b D a , D b a = b a=b a = b A \mathcal{A} A
The metric measures the parameters of the unique normalised representative in the chosen class. Eigenvalues cannot recover Euclidean positions or transformation parameters: translations preserve the spectrum, and taking an infimum of Hausdorff distances over arbitrary transformations of both domains gives zero under small scalings. For a known common transformation T T T ∣ T ′ ∣ |T'| ∣ T ′ ∣
Overlap with algebraic domains
A domain has a real algebraic boundary if its boundary is contained in the zero set of a nonzero polynomial in R [ x , y ] \mathbb{R}[x,y] R [ x , y ]
Proposition C.6 (Overlap with algebraic boundaries). Let Ψ ( w ) = w + ∑ n = 1 m a n w − n \Psi(w)=w+\sum_{n=1}^{m}a_nw^{-n} Ψ ( w ) = w + ∑ n = 1 m a n w − n { ∣ w ∣ ≥ 1 } \{|w|\geq1\} { ∣ w ∣ ≥ 1 } Ψ ′ ( w ) ≠ 0 \Psi'(w)\ne0 Ψ ′ ( w ) = 0 ∣ w ∣ = 1 |w|=1 ∣ w ∣ = 1 Ψ ( { ∣ w ∣ = 1 } ) \Psi(\{|w|=1\}) Ψ ({ ∣ w ∣ = 1 }) F = 0 F=0 F = 0
Proof. For ∣ w ∣ = 1 |w|=1 ∣ w ∣ = 1
z = Ψ ( w ) , ζ = Ψ ( 1 / w ) = z ‾ . z=\Psi(w),\qquad\zeta=\Psi(1/w)=\overline{z}. z = Ψ ( w ) , ζ = Ψ ( 1/ w ) = z . The rational functions z ( w ) , ζ ( w ) z(w),\zeta(w) z ( w ) , ζ ( w ) C ( w ) \mathbb{C}(w) C ( w ) A ( z , ζ ) A(z,\zeta) A ( z , ζ ) z = x + i y z=x+iy z = x + i y ζ = x − i y \zeta=x-iy ζ = x − i y x , y x,y x , y
Our families with prescribed tails impose additional coefficient relations and a cone restriction. For example, the comparable-coefficient m = 2 m=2 m = 2 a 1 a_1 a 1 ∣ P ( x + i y ) ∣ 2 − R 2 = 0 |P(x+iy)|^2-R^2=0 ∣ P ( x + i y ) ∣ 2 − R 2 = 0 ∣ Q ( x + i y ) ∣ 2 − R 2 = 0 |Q(x+iy)|^2-R^2=0 ∣ Q ( x + i y ) ∣ 2 − R 2 = 0
The results in [5 , 6 ] based on finitely many GPTs impose polynomial normalisation, contrast, and nondegeneracy conditions in addition to algebraicity. A prescribed analytic tail need not give an algebraic boundary. Conversely, an algebraic boundary may have an exterior map with infinitely many nonzero Laurent coefficients.
Declaration of AI use
The authors interacted with and guided ChatGPT 6 whilst developing the proofs and then wrote the paper. The authors take full responsibility for the accuracy and integrity of all content in this paper.
