Effective Hamiltonians and non-Hermitian physics

No laboratory system is truly isolated: an atom interacts with vacuum fluctuations, a quantum dot is wired to metallic leads, a spin couples to the phonon bath of its host crystal. One can always insist on isolation by enlarging the system to encompass the environment, but the resulting Hilbert space is then intractably large. What one actually needs is an equation of motion for the relevant degrees of freedom alone.

The Feshbach–Löwdin projection method [1, 2, 3, 4, 5] provides exactly this. Introducing projectors $P$ onto the relevant subspace $\mathcal{H}_L$ and $Q=1-P$ onto the eliminated subspace $\mathcal{H}_R$, the full eigenvalue problem $H|\psi\rangle=E|\psi\rangle$ reduces exactly to an effective problem on $\mathcal{H}_L$:

When the eliminated sector has only a discrete spectrum, $\Sigma(E)$ is real on the real axis and $H_{\mathrm{eff}}$ remains Hermitian. When the eliminated sector contains an open continuum—as for a bath, a radiation field, or a semi-infinite lead—and the energy lies within that continuum, the retarded self-energy generically acquires an imaginary part, provided that the state under study is actually coupled to the open channel; $H_{\mathrm{eff}}$ is then non-Hermitian, with complex poles encoding finite lifetimes. Non-Hermitian physics is therefore not an exotic modification of quantum mechanics: it is the natural outcome of projecting an open system onto the subspace one cares about.

Analytically transparent examples of this chain—from a Hermitian Hamiltonian, through a Green’s function, to an explicit non-Hermitian $H_{\mathrm{eff}}$—are relatively rare. The purpose of this article is to exhibit one, using nothing beyond matching conditions and a Wronskian, and to turn it into a controlled setting in which the interplay between boundary conditions and the Green’s-function formalism becomes fully transparent. Although our setting is deliberately elementary, the boundary-response quantity that emerges from eliminating the right sector is closely analogous in spirit to the surface Green function of a lead, or equivalently to an embedding self-energy in mesoscopic quantum transport: the environment is reduced to an energy-dependent object acting only at the contact [6].

The model is a particle of mass $m$ in a one-dimensional box $[-a/2,a/2]$ with a repulsive delta barrier $+g\delta(x)$ ($g>0$) at its center. On the left half-well $\mathcal{H}_L$, $H_{\mathrm{eff}}$ is the free kinetic operator, with all information about the eliminated sector encoded in a single boundary matching condition

The coefficient $\beta$ is real for a closed well with real $g$, purely imaginary for a purely imaginary coupling, and necessarily complex for any real coupling in an open geometry. The quantum Zeno effect [7, 8] emerges as a direct consequence when $\beta$ is purely imaginary: rather than draining the system faster, a stronger absorber slows the decay and, in the limit of infinite absorption, freezes the dynamics into a perfectly Hermitian one.

The article is organized as follows. Section I recalls the exact spectrum of the model and identifies the even sector as the arena for all non-trivial physics. Section II derives the matching condition by direct elimination of the right half-well: this first approach is elementary but model-specific, and already suffices to exhibit the quantum Zeno effect in Sec. III. To go further, Sec. IV rederives the same result via the Green’s function and Dyson equation; presenting the two derivations in parallel turns the Green’s-function construction into a concrete algebraic manipulation of the underlying wave function rather than an abstract apparatus, recasts $\beta$ in an effective-boundary language closely related to self-energy methods, and provides a framework that does not rely on any closed-form expression for the eliminated half. The power of this reformulation is subsequently demonstrated in Sec. V, where the closed right half-well is replaced by a semi-infinite continuum: the effective boundary term acquires an imaginary part, the boundary parameter becomes complex, and resonances of finite width emerge as the inevitable consequence of an open boundary.

The model and its exact spectrum

We consider a particle of mass $m$ in the symmetric well $[-a/2,a/2]$ with a repulsive delta barrier at the center,

with $g>0$ and $\psi(\pm a/2)=0$. This “particle-in-a-box plus delta” model has a long pedagogical tradition: it has been analyzed as a perturbation problem [11], through its strong- and weak-coupling asymptotics [12], by factorization methods [13], through a variational treatment [14], and more recently with a detailed revisiting of its bound states under various coupling signs [15]. Our perspective here differs: rather than solving the problem within the closed box, we use it as the minimal arena in which projection, effective Hamiltonian, and non-Hermiticity can be exhibited in closed form. The delta barrier imposes continuity of $\psi$ and a derivative jump at $x=0$,

Because the barrier sits at the symmetry center, parity $P:x\mapsto-x$ commutes with $H$. Odd states necessarily vanish at the origin and are completely immune to the barrier; their energies are those of the empty well,

This parity selection rule makes the model pedagogically valuable: the Hilbert space cleanly separates into an unperturbed sector (odd states) and an interacting sector (even states), allowing the projection to be carried out entirely within the even sector. For even states, the left-half ansatz $\phi_L(x)=A\sin(k(x+a/2))/\sin(ka/2)$ with $k=\sqrt{2mE}/\hbar$, normalized so that $\phi_L(0)=A$, yields the exact quantization condition

In the limit $g\to0$, $k\to(2n-1)\pi/a$ (odd-indexed levels of the empty well); for $g\to\infty$, $k\to2n\pi/a$ (even-indexed levels of a half-well of length $a/2$). The even-sector levels thus interpolate continuously between these two limits, illustrating level repulsion.

Let’s introduce the single quantity around which the whole discussion will revolve. On the left half-well the bulk equation is that of a free particle, so an even eigenstate restricted to $[-a/2,0]$ is entirely determined by the Dirichlet condition $\phi_L(-a/2)=0$ together with the value of its logarithmic derivative at the barrier. We therefore introduce the boundary parameter

Once $\beta$ is known, the whole influence of the barrier and of the right-hand sector on the left half-well is encoded in the boundary matching condition $\phi'_L(0)=\beta\phi_L(0)$: nothing else from $x\ge0$ is needed. Evaluating the left ansatz at the origin gives $\beta=k\cot(ka/2)$, so that the exact quantization condition (6) is nothing but the statement $\beta=-mg/\hbar^2$. The parity selection rule ensures that this single number carries all the non-trivial physics: the odd sector, having $\phi(0)=0$, is blind to the barrier and plays no role in what follows.

Effective Hamiltonian by direct elimination

We restrict attention to the left half-well $[-a/2,0]$ and seek an effective operator $H_{\mathrm{eff}}$ acting there, whose eigenvalues coincide with those of $H$ in the even sector. This is the Feshbach–Löwdin construction of (1) in its most elementary incarnation, with $P$ projecting onto $\mathcal{H}_L=L^2([-a/2,0])$ and $Q$ onto $\mathcal{H}_R=L^2([0,a/2])$; since the two halves communicate only through the single point $x=0$, the boundary is where the whole self-energy will end up, and $\beta$ is what remains of it. $\phi_L$ and $\phi_R$ denote the restrictions of one and the same even eigenstate to $[-a/2,0]$ and $[0,a/2]$ respectively; they are two distinct functions, each solving a free Schrödinger equation in its own half-well, and they are related only through the two matching relations holding at $x=0$. We call interface the point $x=0$ at which these two restrictions meet the barrier. The elimination now proceeds in three steps:

Step (i): the eliminated (right) sector. In the right half-well $[0,a/2]$, the solution satisfying the Dirichlet wall at $x=a/2$ is $\phi_R(x)=\phi_R(0)\sin(k(a/2-x))/\sin(ka/2)$. This sector is thus entirely characterized, as far as the interface is concerned, by its own logarithmic derivative

Step (ii): the two matching relations at the interface. The two sectors communicate only through continuity, which transfers the amplitude, and (4), which transfers the slope: $\phi_R(0)=\phi_L(0)\equiv A$ and $\phi'_R(0^+)-\phi'_L(0^-)=2mgA/\hbar^2$.

Step (iii): elimination of the right-hand data. Combining the previous equations and isolating the left-hand data $\phi'_L(0^-)$ gives the energy-dependent boundary matching condition

$\phi_L'(0)=\beta_{\mathrm{eff}}(E)\,\phi_L(0),\qquad\beta_{\mathrm{eff}}(E)=-k\cot\left(\frac{ka}{2}\right)-\frac{2mg}{\hbar^2}.$ {#eq:9}

The free solution in the left half-well independently satisfies $\frac{\phi_L'(0)}{\phi_L(0)}=k\cot\left(\frac{ka}{2}\right)$, and requiring this logarithmic derivative to coincide with the boundary condition imposed by the eliminated sector, $\beta=\beta_{\mathrm{eff}}(E)$, gives

that is, the exact quantization condition (6). The eigenvalues of $H$ in the even sector are thus recovered as the energies at which the effective boundary condition is compatible with the left-hand dynamics—which is exactly what a Feshbach–Löwdin elimination is meant to deliver.

On shell, i.e. on a solution $E_n$ of Eq. (9), the boundary parameter takes the energy-independent value

with $\beta$ the boundary parameter of Eq. (7). Off shell, the boundary term generated by the elimination is energy dependent, as Eq. (9) makes explicit.

The even eigenstates of $H$, restricted to $[-a/2,0]$, therefore satisfy $H_{\mathrm{eff}}\phi_L=E\phi_L$ with $x\in[-a/2,0]$, where

with boundary conditions $\phi_L(-a/2)=0$ and Eq. (10). The bulk operator is the free kinetic energy; all physics of the barrier and the eliminated half-well is encoded solely in $\beta$.

Two features of this construction deserve emphasis. First, the on-shell value $\beta=-mg/\hbar^2$ is energy-independent, whereas the boundary term $\beta_{\mathrm{eff}}(E)$ produced by the elimination is not. The cancellation of the $k$-dependence occurs only on the solutions of Eq. (9), and it occurs there only because the delta sits exactly at the midpoint, making the left and right propagators carry the same $k\cot(ka/2)$ factor. For a delta at a generic position, or for a more complex scatterer, the on-shell value of $\beta$ would depend on $E$ and the eigenvalue problem would become nonlinear, requiring a self-consistent solution; the reflection symmetry of the present model is what makes the answer a single number. Second, the criterion for self-adjointness is entirely controlled by $\operatorname{Im}(\beta)$: as shown in Appendix A, the defect $\langle\phi|H_{\mathrm{eff}}\psi\rangle-\langle H_{\mathrm{eff}}\phi|\psi\rangle$ reduces to a boundary term proportional to $\operatorname{Im}(\beta)$, which vanishes for real $g$ and is nonzero otherwise. Projecting a Hermitian system onto a subsystem therefore need not break self-adjointness; the outcome is determined solely by whether $\beta$ is real or complex.

Strong imaginary coupling: the quantum Zeno effect

The previous section showed that $H_{\mathrm{eff}}$ is self-adjoint if and only if $\beta$ is real. What, then, does a complex $\beta$ describe? The simplest way to produce one is to make the coupling itself imaginary, $g\to-i\gamma$ with $\gamma>0$ (a sink, not a source), so that the barrier becomes the anti-Hermitian potential $-i\gamma\,\delta(x)$ in Eq. (3). Such a term is an absorber: a point-like channel at which probability leaves the system and is not returned. It is no ad hoc device, but itself the outcome of an elimination—the particle is coupled at $x=0$ to degrees of freedom (photon modes of a resonant beam, detector states, an external continuum) that are never observed again, and eliminating them in the sense of Eq. (1) leaves a self-energy which, for a fast and structureless (Markovian) channel, reduces to a purely imaginary, energy-independent term at the coupling point. As shown in Appendix A, $H_{\mathrm{eff}}$ then generates the no-jump (null-measurement) evolution of a monitored system, the missing norm being the probability that the particle has been absorbed [28, 29]. For the full box with the imaginary delta potential, writing $i\hbar\partial_t\psi=(H_0-i\gamma\delta(x))\psi$ and differentiating the total norm gives

The corresponding statement for the effective problem on the left half-well alone follows from the boundary term computed in Appendix A: with $N_L=\int_{-a/2}^{0}|\psi_L|^2dx$,

the factor of two between the two expressions being simply the second half-well, which loses probability at the same rate. The probability is therefore lost only in proportion to the density at the absorber. An eigenstate of $H_{\mathrm{eff}}$ has a complex energy $E_n$ and evolves as $e^{-iE_nt/\hbar}$, so its norm decays exponentially, $|\psi_n(t)|^2\sim e^{-\Gamma_nt}$, with the decay rate $\Gamma_n=-2\operatorname{Im}(E_n)/\hbar\geq0$, whose inverse is the lifetime of the mode.

The paradox: a stronger absorber decays more slowly

We now consider a purely imaginary coupling $g=-i\gamma$ with $\gamma>0$ (a pure absorber at $x=0$). The boundary parameter becomes purely imaginary, $\beta=i m\gamma/\hbar^2$, and the even-sector quantization condition reads

All even eigenenergies are complex with $\operatorname{Im}(E_n)<0$: every even mode decays. A naive expectation is that a stronger absorber produces a larger imaginary part; the exact solution shows the opposite. A very strong absorber forces any eigenstate to have nearly zero amplitude at $x=0$, but a state with exactly zero amplitude cannot be absorbed.

For large $\gamma$, writing $k_n=2n\pi/a+\delta_n$ and expanding gives $\delta_n\simeq-4n\pi i\hbar^2/(m\gamma a^2)$, so the decay rate of even mode $n$ is

This is the Zeno decay rate: $\Gamma_n\propto1/\gamma$ decreases as the absorber strengthens. Figure 1 confirms this behavior: panel (a) shows the real parts migrating toward the nearest odd-sector level as $\gamma$ grows, reflecting the effective Dirichlet condition imposed by the strong absorber, while panel (b) confirms the $1/\gamma$ suppression.

As $\gamma\to\infty$, the matching condition transitions to Dirichlet ($\phi_L(0)=0$), and $H_{\mathrm{eff}}$ becomes the operator of an isolated half-well: perfectly Hermitian. The non-Hermitian dynamics self-projects onto the Hermitian Zeno subspace—the space of states that avoid the absorber. This is the quantum Zeno effect [7, 8]: frequent measurement (here, continuous coupling to a loss channel) confines the system to the dark subspace the measurement cannot access.

The condition $\phi(0)=0$ selects the domain of $H_{\mathrm{eff}}$: as $\gamma$ grows, that domain approaches the Dirichlet one at $x=0$, and the long-lived dynamics is generated by the Hermitian Hamiltonian of two disconnected half-wells—the Zeno-subspace structure of Refs. [8, 9]. Components strongly coupled to the loss channel disappear during an initial transient, but the widths of the survivors decrease as $1/\gamma$: the law of Eq. (15) is the competition between the two factors of Eq. (13), the loss rate being the product of $\gamma$, which grows, and of $|\phi(0)|^2$, which the same $\gamma$ suppresses as $1/\gamma^2$. The absorber excludes probability from its own location until nothing is left to absorb; the effect was observed on a trapped-ion transition [10].

To illustrate the dynamics, we choose as initial state the ground state of the isolated left half-well,

Because this state is entirely localized in the left half-well, its odd and even parity components have equal norm. Moreover, its odd component is exactly proportional to the first odd eigenstate of the full box, $\phi_1^{(-)}(x)\propto\sin(2\pi x/a)$. In the strong-absorption regime, the long-time decay of the even component is dominated by the lowest even resonance, hence governed by the Zeno timescale $\tau_Z\equiv\Gamma_1^{-1}=m^2\gamma a^3/(16\pi^2\hbar^3)$, whereas the odd component is immune to the absorber. Hence, for $t\gg\tau_Z$, the dynamics is entirely governed by $\phi_1^{(-)}$, and the long-time probability balance is

where the first term is the absorbed fraction and the last two terms are the asymptotic probabilities in the left and right half-wells (see Figure 2). Note that the two limits do not commute, $\lim_{\gamma\to\infty}\lim_{t\to\infty}\ne\lim_{t\to\infty}\lim_{\gamma\to\infty}$. For fixed finite $\gamma$, waiting until $t\gg\tau_Z$ removes the even component and leaves the dark odd one, with the balance above. At fixed observation time, $\gamma\to\infty$ sends $\tau_Z\to\infty$ and gives unitary dynamics in the two disconnected half-wells: the particle stays in the left well. Both orders are physical—a fixed apparatus observed for a long time, versus a freezing of the dynamics by an ever stronger measurement.

The Green's function formalism

The derivation of Sec. II relies on an explicit closed-form expression for $\phi_R$. A more general approach uses the Green’s function $G_R(0,0;E)$—the single quantity characterizing the response of the right half at the interface—which carries over unchanged when the right half is replaced by an arbitrary environment [16, 17, 18].

The result of Sec. II is already exact, and the formalism that follows will, at first sight, only rederive it. Its advantage is modularity. The elimination of Sec. II used the explicit sine of the closed right half-well and would have to be redone from scratch for any other environment, whereas the Green’s-function route stores the eliminated sector in a single scalar function of the energy, its boundary response $G_R^{-1}(E)$, and gives $\beta_{\mathrm{eff}} \propto-g-G_R^{-1}(E)$. Replacing the closed box by a lead, a bath, or a structured continuum then amounts to substituting one formula for $G_R^{-1}$ by another, everything else being unchanged—which is precisely what is done in Sec. V. Two further benefits follow. First, $G_R^{-1}(E)$ provides the boundary representation of the embedding self-energy of the Feshbach–Löwdin construction, Eq. (1): because the spatial sectors of a differential operator communicate through interface conditions rather than through an ordinary matrix element, the self-energy appears here not as an additional bulk potential but as an energy-dependent boundary term, so that the correspondence with $H_{LR}(E-H_{RR})^{-1}H_{RL}$ is one of role rather than of literal identity. Second, an environment with a continuous spectrum makes $G_R^{-1}$ complex, so that non-Hermiticity is read off directly from the analytic structure of the response, without any need to postulate a complex coupling. This mechanism underpins Fano resonances in quantum transport [19, 20], Anderson impurity physics [21, 22], radiative linewidths in cavity QED [23, 24], and quasiparticle damping in metals [25, 26].

Green's function and Dyson equation

For $E$ outside the spectrum of $H_0$, the resolvent kernel $G_0(x,x';E)$ is defined by $(E-H_0)G_0(x,x';E)=\delta(x-x')$. It is meromorphic in $E$, with simple poles at the eigenvalues of $H_0$, and on an interval with Dirichlet walls admits the Wronskian representation (see Appendix § 1.1.2)

where $u_1$ ($u_2$) vanishes at the left (right) wall. Evaluating at $x=x'=0$ for the free box with $u_1(x)=\sin k(x+a/2)$ and $u_2(x)=\sin k(a/2-x)$, gives

This quantity diverges at the even levels of the empty well (antinodes at $x=0$) and vanishes at the odd levels (nodes at $x=0)—a direct signature of the parity immunity noted in Sec. I.

For the Hamiltonian (3), the full Green’s function satisfies the Dyson equation (see Appendix C)

Because $V=g\delta(x)$ is a rank-1 perturbation, setting $x=x'=0$ closes the equation algebraically:

the exact resummation of the Born series. The even-sector eigenvalues are the poles of this expression, i.e. the solutions of $G_0^{-1}(0,0;E)=g$, which reproduces Eq. (6) upon substituting Eq. (19).

Effective boundary term and boundary matching condition

To generalize beyond the closed well, we express $G_0(0,0;E)$ in terms of the separate responses of the two half-wells. Defining the logarithmic derivatives $\Lambda_L\equiv u'_L(0)/u_L(0)$ and $\Lambda_R\equiv-u'_R(0)/u_R(0)$ (with sign convention following the outward normal from each half-well), the symmetry $u_R(x)=u_L(-x)$ gives $\Lambda_L=\Lambda_R=k\cot(ka/2)$. Evaluating Eq. (18) at the interface and factoring out $u_L(0)u_R(0)$ from the Wronskian gives $G_0(0,0;E)=-(2m/\hbar^2)/(\Lambda_L+\Lambda_R)$ so that, with $G_{L,R}^{-1}\equiv(\hbar^2/2m)\Lambda_{L,R}$, one obtains

where $G_{L,R}^{-1}(E)\equiv(\hbar^2/2m)k\cot(ka/2)$. This identity, which expresses the full interface propagator as the series combination of two half-well propagators, is the Green’s-function counterpart of the direct elimination performed in Sec. II. The secular equation $G_0^{-1}(0,0;E)=g$ then reads $G_L^{-1}(E)+G_R^{-1}(E)=-g$. Isolating the left half-well defines an effective boundary term acting on the left half-well,

Using $G_L^{-1}=(\hbar^2/2m)\phi'_L(0)/\phi_L(0)$ and $G_R^{-1}=(\hbar^2/2m)k\cot(ka/2)=-g/2$ on a solution of the secular equation, one recovers directly $\beta=-mg/\hbar^2$, confirming that the two approaches are equivalent.

The modularity announced at the beginning of this section is now manifest in Eq. (23): the barrier contributes $-g$, the eliminated sector contributes $-G_R^{-1}(E)$, and these are the only two ingredients of the effective boundary term $\Sigma_R(E)$ felt by the left half-well. Nothing has been gained in accuracy—the two routes are compared line by line in Table I—but the right half-well now sits in one clearly identified slot. If its content has a continuous spectrum, $G_R^{-1}$ becomes complex, $\Sigma_R$ acquires an imaginary part, and $H_{\mathrm{eff}}$ is non-Hermitian even for real $g$: this is the subject of the next section.

Open boundary: a continuum as the discarded subspace

We now remove the right wall while keeping the same projection strategy as in Sec. IV. The particle is still confined by an infinite wall at $x=-a/2$ and encounters the real barrier $g\,\delta(x)$ at the interface $x=0$, but the region $x>0$ is now an open continuum. We keep $\mathcal{H}_L=L^2([-a/2,0])$ as the subspace of interest [27], and continue to denote by $G_R$ the boundary response of the eliminated right sector.

In the present open geometry, the right sector is the free half-line $[0,+\infty)$ with outgoing boundary conditions. Its Dirichlet-to-Neumann map is simply

so that, with the convention of Sec. IV,

Unlike the closed half-well response, which is real and inherits the pole structure of a discrete spectrum, the open right sector contributes a purely imaginary boundary term: it represents an irreversible escape channel rather than a set of bound modes.

The boundary matching relation derived in Sec. IV remains unchanged, $G_L^{-1}(E)+G_R^{-1}(E)=-g$. Substituting Eq. (25) gives

Using $G_L^{-1}(E)=\hbar^2\phi'_L(0)/(2m\phi_L(0))$, we obtain the effective matching condition on the left half-well,

The real part $-2mg/\hbar^2$ is the contribution of the delta barrier, whereas the imaginary term $+ik$ is the fingerprint of the continuum. Since $\operatorname{Im}\beta(E)=k\ne0$ for real $k>0$, the effective Hamiltonian on $[-a/2,0]$ is non-Hermitian for any real $g$: here non-Hermiticity is not a property of the coupling itself, but the generic hallmark of an open boundary: it survives however weak the coupling, as long as the continuum is accessible. Applying Eq. (27) to the left-well ansatz $\phi_L(x)=A\sin k(x+a/2)$ yields the resonance condition

Its complex roots $k_n$ determine the resonance energies $E_n=\hbar^2k_n^2/(2m)$; their imaginary parts encode the leakage of probability from the left well into the continuum. For real $g$, one finds $\operatorname{Im}(k_n)<0$, hence $\operatorname{Im}(E_n)<0$. The

decay rate $\Gamma_n$ measures the probability loss into the continuum. An alternative derivation of Eq. (28) is detailed in Appendix D.

The coupling to the continuum also displaces the real part of $E_n$ away from the isolated half-well value, providing the reactive counterpart of the decay width. In this sense, the resonance pole carries both a dissipative component (the width) and a dispersive one (the frequency shift), much like the linewidth and level renormalization generated by a self-energy in open quantum systems. In more structured environments this dispersive contribution appears explicitly as the real part of the self-energy; in the present flat half-line model it is encoded implicitly in the position of the complex pole itself.

The origin of the decay is transparent at the level of the boundary condition itself: the positive imaginary part of $\beta(E)$ in Eq. (27) is the signature of an outgoing flux through the interface at $x=0$. Thus, the complex poles of Eq. (28) represent genuine resonances: probability is not trapped in the left well, but leaks irreversibly into the continuum. This leakage vanishes in two physically distinct limits. First, if the wavefunction amplitude at the interface tends to zero, the state decouples from the continuum. Second, near threshold $k\to0$, the group velocity in the continuum vanishes, so escape is also suppressed. The first mechanism is precisely the one at work in the Zeno regime discussed in Sec. III: a strong absorber expels probability from the interface, driving $|\phi_n(0)|\to0$ and thereby reducing, rather than enhancing, the decay.

Summary and outlook

A single scalar parameter—the boundary parameter $\beta$ defined by $\phi'_L(0)=\beta\phi_L(0)$—unifies four physical regimes of the same model: Hermitian dynamics, non-Hermitian decay by design, resonances induced by openness, and the quantum Zeno effect. In every case the bulk operator $H_{\mathrm{eff}}=-(\hbar^2/2m)\partial_x^2$ is unchanged; all the physics of the eliminated sector is encoded solely in $\beta$. Two lessons stand out. First, projection reshapes boundary conditions, not bulk dynamics. Second, non-Hermiticity arising from openness requires no fine-tuning: whenever the discarded subspace contains an accessible continuum to which the state is actually coupled, and outgoing conditions are imposed, the boundary self-energy acquires an imaginary part at real energies and $H_{\mathrm{eff}}$ becomes non-Hermitian regardless of the coupling strength.

According to Eq. (1), eliminating the $Q$-sector produces an effective Hamiltonian $H_{\mathrm{eff}}(E)=H_{LL}+\Sigma(E)$ on the $P$-sector alone, with all the eliminated physics stored in the self-energy $\Sigma(E)=H_{LR}(E-H_{RR})^{-1}H_{RL}$. Here every element of it is explicit. The $P$-sector is $L^2([-a/2,0])$ and the $Q$-sector is whatever lives at $x\ge0$; the coupling $H_{LR}$ is supported at the single point $x=0$—so that the two sectors communicate through interface conditions rather than through an ordinary matrix element—which is why the eliminated sector does not renormalize the bulk operator $H_{LL} = -(\hbar^2/2m)\partial_x^2$ at all, but leaves an energy-dependent boundary term; the resolvent $(E-H_{RR})^{-1}$ of the eliminated sector becomes its boundary response $G_R(E)$, computed in Sec. IV for a closed half-well and in Sec. V for a continuum; and the boundary representation of the embedding self-energy is $\Sigma_R(E) = -g - G_R^{-1}(E)$, which is $\beta_{\mathrm{eff}}(E)$ up to the factor $\hbar^2/2m$. Comparing Eq. (10) with Eq. (27) then exhibits the mechanism stated below Eq. (1) in its simplest setting: the closed half-well gives $\beta= -mg/\hbar^2 \in\mathbb{R}$, whereas the open half-line adds the term $+ik$ inherited from the continuum, whose imaginary part is the decay width. The delta barrier in a box is thus a Feshbach–Löwdin projection carried out by hand from beginning to end.

The two derivations of $\beta$—direct elimination and the Green’s-function route—converge to the same result, and it is precisely the redundancy that is instructive. By deriving $\beta$ twice, in two languages, one sees each step of the Green’s-function construction reflected in an explicit algebraic manipulation of the underlying wave function. Table I summarizes the correspondence: the two columns differ not in their content but in their extensibility.

Here the elementary approach builds intuition, while the boundary-response formulation opens the door to arbitrary environments: a structured spectral density would introduce memory effects and non-Markovian dynamics [28], making $\beta$ frequency-dependent; promoting the model to a many-body setting—for instance a Tonks–Girardeau gas in the left half-well—would connect the boundary-matching physics to correlated open quantum systems [29].

Acknowledgments

D.G.-O. acknowledges support from the Institut Universitaire de France.

Author contributions

D.G.-O. conceived the study and led the investigation. U.L. performed the formal analysis, validated the results, and critically revised the manuscript.

Data availability

No new data were created or analysed in this study. All results are reproducible from the equations provided in the manuscript.

Self-adjointness and non-Hermiticity of $H_{\mathrm{eff}}$

For any two functions $\phi,\psi$ satisfying the same Dirichlet condition at $x=-a/2$ and the same matching condition $\phi'(0)=\beta\phi(0)$ at $x=0$, integration by parts gives

so that $H_{\mathrm{eff}}$ is self-adjoint if and only if $\operatorname{Im}(\beta)=0$. Real $g$ gives $\beta=-mg/\hbar^2\in\mathbb{R}$, whereas $g=-i\gamma$ ($\gamma>0$) gives $\operatorname{Im}(\beta)=m\gamma/\hbar^2$ and a departure from self-adjointness entirely localized at $x=0—as it must be, the defect of any Sturm–Liouville operator being a boundary term [30, 31].

The resulting term $-i\gamma|\phi(0)|^2$ is the precursor of the Lindblad description of a local particle-loss channel: the imaginary potential generates the no-jump evolution alone, the jumps being described by $\hat{L}=\sqrt{2\gamma/\hbar}\,\hat{\psi}(0)$, which removes the particle rather than projecting it at $x=0$, and which reproduces the loss term of Eq. (3) through $-(i\hbar/2)\hat{L}^{\dagger}\hat{L}$.

Wronskian representation of the Green’s function

Let $u_1,u_2$ be two linearly independent solutions of $(E-H_0)u=0$ satisfying the left and right boundary conditions respectively. For $x\ne x'$, the Green’s function must take the form $G=Au_1(x)$ for $x<x'$ and $G=Bu_2(x)$ for $x>x'$. Continuity at $x'$ gives $Au_1(x')=Bu_2(x')$, and integrating $(E-H_0)G=\delta(x-x')$ across $x'$ gives the jump condition $\partial_xG|_{x'+}-\partial_xG|_{x'-}=2m/\hbar^2$. Solving for $A$ and $B$ yields

where $x_< = \min(x,x')$, $x_> = \max(x,x')$, and $W[u_1,u_2]=u_1u_2'-u_1'u_2$ is the (constant) Wronskian.

Dyson equation from first principles

Let $G_0=(E-H_0)^{-1}$ and $G=(E-H_0-V)^{-1}$. Writing $E-H_0=(E-H_0-V)+V$, multiplying left by $G_0$ and right by $G$ gives $G=G_0+G_0VG$. In position representation with $V(y)=g\delta(y)$:

Setting $x=x'=0$ closes the equation at the single point $(0,0)$ and yields $G(0,0;E)=G_0(0,0;E)/[1-gG_0(0,0;E)]$, the exact resummation of the Born series.

	Direct elimination (Sec. II)	Boundary response (Sec. IV)
Object describing the eliminated sector	closed-form $\phi_R$, (8)	response $G_R^{-1}(E)$, (22)
Tool used at $x=0$	continuity + jump condition	Dyson equation, rank-1 closure
Result of the elimination	$\beta_{\mathrm{eff}}(E)=-2mg/\hbar^2-k\cot(ka/2)$, eq:9	$\Sigma_R(E)=-g-G_R^{-1}(E)$, (23)
On-shell value	$\beta=-mg/\hbar^2$	$\beta=-mg/\hbar^2$ (same)
Prerequisite	the right sector must be solvable in closed form	only $G_R^{-1}(E)$ is needed
Extension to a continuum	possible but must be redone (Appendix D)	substitute $G_R^{-1}=-i\hbar^2 k/2m$

Direct derivation of the open-boundary matching condition

For completeness, we briefly rederive the effective boundary condition of the open geometry without using Green’s functions. In the right region $x>0$, the outgoing solution at energy $E=\hbar^2k^2/(2m)$ is $\phi_R(x)=C e^{ikx}$. Continuity at the interface gives $C=\phi_L(0)$, hence

Using the jump condition across the delta barrier, we obtain

which is (27) of the main text. Finally, inserting the left-well form $\phi_L(x)=A\sin(k(x+a/2))$ gives (28) independently of the Green-function formalism.