Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states

Bj\"orn Garbrecht; Nils Wagner

REVIEW 3 major objections 4 minor 3 cited by

Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives an exact equality between path integrals on a complexified contour and spectral sums over generalized eigenvalues, and uses it to show that instanton decay rates and resonant-state decay rates are two representations of…

desk verdict A serious, transparent formal paper that plausibly links resonant-state boundary conditions to the instanton contour, but its advertised one-to-one correspondence rests on an explicitly admitted conjecture. read the letter →

arxiv 2507.23125 v1 pith:KW7NTGP7 submitted 2025-07-30 hep-th math-phmath.MPquant-ph

Björn Garbrecht , Nils Wagner This is my paper

classification hep-thmath-phmath.MPquant-ph PACS 03.65.-w03.65.Xp

keywords complexcontourpathintegralsGamow–SiegertboundaryconditionsStokessectorsfalsevacuumdecayinstantonsresonantstatesPT-symmetricquantummechanicsfunctionalmethods

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the path integral formalism to Schrödinger eigenvalue problems whose boundary conditions are imposed in angular wedges of the complex plane, not on the real axis. Its central result is an exact identity between the propagator written as a contour path integral and the same propagator written as a sum over generalized eigenvalues, together with the trace version of that identity. Applied to a metastable potential, the identity produces the resonant ground-state energy from the false-vacuum and bounce contributions, which is the precise statement behind the instanton method's heuristic contour deformation. If correct, the Euclidean instanton calculation and the resonant-state calculation of decay rates are two views of one boundary value problem, and the same functional machinery applies to other sectorial spectral problems such as PT-symmetric ones.

What carries the argument

The load-bearing objects are the Stokes sectors of the polynomial potential, the angular wedges in which a solution is exponentially subdominant, and the complex contour $\Gamma$ that joins two chosen sectors and carries the functional integral. On this contour the eigenvalue problem becomes a non-Hermitian Hamiltonian on the real line, whose spectral resolution requires a bi-orthogonal basis; the paper shows the dual basis is simply $\gamma'(s)$ times the original wave function, which makes the propagator independent of the parametrization. The path integral side is built from short-time propagators whose position-dependent kinetic terms require operator-ordering substitution rules, and the continuum limit yields an integration over maps into $\Gamma$. The identity that carries the argument is the master formula (3.34), whose trace version (3.37) serves as a generalized partition function.

What would settle it

Compute the two sides of the master formula (3.34) independently for a concrete polynomial potential with finite $T$: evaluate the path integral on a discretized contour $\Gamma$ and sum the first many generalized eigenvalues with their $\Gamma$-normalizations; a disagreement would refute the exact identity, while agreement would test the contour assumptions. For the tunneling application, compare the predicted resonant energy from $E_0^{(\mathrm{resonant})} = -\hbar \lim_{T\to\infty} T^{-1} \log[K_{\mathrm{FV}} + \frac{1}{2} K_{\mathrm{bounce}}]$ with the resonant eigenvalue obtained by a direct Wronskian or ODE computation; if additional saddle points contribute, the two results will differ by nonperturbative terms.

Watch

Extended reading notes

Core claim

For a Schrödinger-type operator with polynomial potential whose eigenfunctions are required to decay in two non-adjacent Stokes sectors, the paper derives an exact equality between two representations of the analytically continued propagator: a sum over the generalized eigenvalues weighted by their contour-normalized wave functions, and a path integral whose paths live on a complex contour $\Gamma$ terminating in those sectors. Equating the two gives the master formula (3.34) and its trace form (3.37). In the tunneling case, projecting the Euclidean false-vacuum-to-false-vacuum propagator onto the resonant ground state yields $E_0^{(\mathrm{resonant})} = -\hbar \lim_{T\to\infty} T^{-1} \log[K_{\mathrm{FV}} + \frac{1}{2} K_{\mathrm{bounce}}]$, with the factor $1/2$ arising from the overlap of the false-vacuum and bounce integration cycles and the shot-like saddle absent because the deformed potential is unbounded. This is presented as the first derivation of the instanton prescription from the outgoing Gamow–Siegert boundary conditions of decaying states.

Load-bearing premise

In the tunneling application, the load-bearing premise is that the infinite-dimensional integration contour receives contributions from exactly two saddle points, the false-vacuum trajectory and the bounce, with the bounce entering with overlap factor $1/2$; the authors call this a well-motivated conjecture in section 4.3 because a full decomposition is far out of reach.

Editorial extensions

If this is right

The classic instanton bounce computation and the resonant-state computation of a decay rate become the same boundary value problem, so agreement between the two is a consequence of the spectral identity rather than a coincidence.
The empirically known ingredients of the instanton method, including the factor $1/2$ on the bounce and the omission of shot-like trajectories, are consequences of choosing outgoing boundary conditions, and the sign of the imaginary part selects resonant versus anti-resonant states.
The master formulas give a functional method for computing spectra of nonstandard Schrödinger-type problems, including $\mathcal{PT}$-symmetric ones, without first solving the differential equation.
Excited-state decay rates can be extracted by inserting the corresponding resonant wave functions into the path integral, as formalized by the projection formula (4.11).
The contour constraints in Appendix C delimit the admissible Wick-rotation angles and integration contours, providing a concrete consistency condition for any future application of the method.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct numerical check of (3.34) on a finite polynomial potential, comparing a discretized path integral on $\Gamma$ with the truncated spectral sum, would isolate where the complex-contour construction holds and where the conjectured saddle-point decomposition starts to matter.
If the full decomposition of the integration contour turns out to contain contributions beyond the false-vacuum and bounce saddles, the exact resonant-instanton correspondence would likely survive at leading semiclassical order but fail at nonperturbative level, making decay rates genuinely dependent on the deformation of the potential.
The same contour machinery could be applied to field-theoretic tunneling by replacing the single complex variable with a complexified field configuration space, provided the sectorial boundary conditions can be identified in that setting.
The disjoint-overlap case discussed in Appendix C, where two inequivalent contours terminate in the same sector, could yield vanishing-cycle identities and sharpen the conditions under which the path integral is independent of the contour choice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a serious, carefully written hep-th paper. It derives exact master formulas (3.34)/(3.37) equating complex-contour path integrals to spectral sums for Schrödinger-type eigenvalue problems with sectorial boundary conditions, then applies them to quantum tunneling, obtaining E0^(res) = -ℏ lim_{T→∞} T^{-1} log[K_FV + (1/2)K_bounce]. That last step is the advertised explanation of Callan–Coleman, and it is not proven—the authors themselves label the needed thimble decomposition a 'well-motivated conjecture' (footnote 21). So if you read the abstract, lower your expectations: the 'previously elusive one-to-one correspondence' is a reinterpretation of the instanton method, not a derivation from first principles.

What is genuinely new and good: the master formulas themselves are new, and the path integral derivation is unusually careful. The treatment of the non-Hermitian effective Hamiltonian via bi-orthogonal bases, the detailed handling of short-time propagator ordering ambiguities (Graham/DeWitt substitution rules), and appendices A–D are valuable. The authors are also honest about what is not known: footnote 8 admits the infinite-dimensional completeness assumption is not rigorous; footnote 21 admits the thimble decomposition is beyond reach; footnote 22 notes nonperturbative corrections depend on the deformation, so the correspondence is semiclassical in character. That transparency is a real strength.

The soft spots are exactly where the stress-test puts them. The step from (3.34) to (4.9) requires C[0,T],Γ ≡ J_FV + J_bounce with exact 1/2 overlap; no intersection-number computation is given, and the figure is called an over-simplification. If other thimbles contribute, the equality fails. This is load-bearing, not cosmetic. The biorthogonal completeness concern is less decisive—for sectorial polynomial problems it should follow from Sibuya-type results, and the late-time projection mostly needs the dominant eigenstate. So the core of the paper is a plausible framework plus an open problem, not a closed theorem.

Who is this for? People working on instanton methods, decay rates, or PT-symmetric quantum mechanics. They will find the master formulas and the critique of Andreassen et al. (the T vs T' Wick-rotation mistake) useful. It deserves a serious referee: it is important enough and formally grounded enough to warrant expert scrutiny, even though the central claim will need heavy revision or qualification.

Recommendation: send it to review, with a referee who knows Picard–Lefschetz theory and can push on the thimble decomposition. The authors should be asked to state clearly that the final correspondence is conditional on that conjecture.

Referee Report

3 major / 4 minor

Summary. The paper develops a path integral formalism for Schrödinger-type eigenvalue problems whose boundary conditions are imposed in angular sectors of the complex plane, thereby covering resonant (Gamow–Siegert) and PT-symmetric-type spectra. For a complex contour Γ terminating in the chosen Stokes sectors, the authors define a propagator K_{γ,θ}, derive its spectral representation via a bi-orthogonal basis, and obtain a path integral representation over the function space C([0,T],Γ). Equating the two gives the master formulas (3.34) and (3.37). The paper then applies these formulas to quantum tunneling, arguing that the Euclidean false-vacuum-to-false-vacuum propagator for the deformed, unbounded potential decomposes as C([0,T],Γ) ≡ J_FV + J_bounce with overlap factor 1/2, yielding the resonant ground-state energy E_0^{(resonant)} = -ℏ lim_{T→∞} T^{-1} log[K_FV + (1/2) K_bounce] in (4.9). This is presented as the rigorous explanation of Callan and Coleman's instanton prescription. The paper also contains a critical assessment of the direct method, the steadyon picture, and other approaches.

Significance. If the central equality (3.34) and its trace version (3.37) are established, the paper provides a substantial unification: Euclidean instanton calculus and resonant-state computations become two representations of the same boundary value problem. The derivation is parameter-free and the path integral technology in Appendices A–C is developed in unusual detail, including operator-ordering issues and substitution rules. The authors are also commendably explicit about the limits of their arguments, conceding in footnote 21 and in Section 4.3 that the key thimble decomposition is a conjecture and that the full Picard–Lefschetz structure is out of reach. However, the advertised 'previously elusive one-to-one correspondence' is exactly what rests on that conjectural step, so the significance of the paper is conditional: the master formulas are a genuine technical contribution, but the tunneling application as stated is not yet a proof.

major comments (3)

[§4.3, Eqs. (4.8)–(4.9), and footnote 21] The central application relies on the infinite-dimensional steepest-descent decomposition C([0,T],Γ) ≡ J_FV + J_bounce with overlap factor 1/2. The authors explicitly state that this is 'merely a well-motivated conjecture' and that the full thimble decomposition in the complexified function space is 'far out of reach'. No intersection-number computation is given, and the figure illustrating the decomposition is labeled an 'over-simplified representation'. If any other thimble contributes, or if the overlap is not exactly 1/2, equation (4.9) does not follow and the claimed one-to-one correspondence between instanton and resonant-state decay rates is not established. The manuscript should either prove or substantially support this step, for example by analyzing finite-dimensional truncations or a solvable model, or should explicitly downgrade the conclusion to a conjectural equivalence.
[§3.1, Eq. (3.9), and footnote 8] The spectral representation assumes that the bi-orthogonal eigenfunctions of H_γ and H_γ† form a complete basis in L²(R) with the resolution of identity (3.9), but footnote 8 concedes that the rigorous statement is only finite-dimensional. Since (3.34) and (3.37) are claimed as exact equalities, this missing infinite-dimensional completeness is load-bearing. The authors should either provide a proof or a precise citation for the sectorial polynomial class (for example from the Sibuya–Shin results cited in Appendix D), or state clearly that the master formulas are derived under this unproven assumption.
[§3.2.1–§3.2.3 and Appendix C.1] The derivation of the path integral representation (3.31) requires the local slope constraint (C.1) on the contour Γ, which restricts admissible contours and excludes loops or sharp U-turns. Appendix C.1 admits that contours violating (C.1) can only be rescued by a pinching procedure that is described as an ad hoc argument, with a rigorous extension left as a conjecture. For the resonant-state application, where Γ must end in the intersection S±∩(S0∪...∪Sn), the existence of a representative satisfying (C.1) is not demonstrated. This leaves the derivation of the master formula incomplete for a general contour of the type used in Section 4.3. Please clarify the status of (C.1) for the resonant application and either prove existence of an admissible representative or restrict the master formulas to contours for which the derivation is valid.

minor comments (4)

[§4.3, Figure 12] The caption of Figure 12 states that the thimble decomposition is an 'over-simplified representation'. Because this figure carries the central argument of the tunneling application, a more detailed schematic or an accompanying finite-dimensional analogue would help the reader distinguish what is established from what is conjectured.
[§3.3 and §2] The text contains several typographical artifacts, including 'exactexactexact' in Section 3.3 and 'time-ininininininininininininininininindependent' in Section 2; these should be corrected.
[§3.2, after Eq. (3.22)] The notation '.=' for path-integral equivalence is used in the main text before it is introduced properly in Appendix A; a short parenthetical definition at first use would improve readability.
[§4.1 and §4.3] The term 'thimble' is used somewhat loosely for steepest-descent contours; the authors should define it explicitly in the main text, since standard Picard–Lefschetz terminology distinguishes thimbles from the original integration cycles.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the master formula is a derived identity between two representations of the same contour propagator, and the instanton application rests on an admitted thimble-decomposition conjecture, which is a completeness gap rather than a circular input.

full rationale

The paper's central equality (3.34)/(3.37) is obtained by equating the spectral representation (3.11) and the path-integral representation (3.31) of the same contour propagator K_gamma,theta. Both representations are derived from the same generalized eigenvalue problem (3.1); the equality is therefore a consistency statement, not a fitted prediction, and no parameter is adjusted to force agreement. The application to tunneling uses (4.7) with the spectral representation to define the resonant ground-state energy, and then posits the thimble decomposition C([0,T],Gamma) = J_FV + J_bounce in (4.8). This is the load-bearing step for the advertised one-to-one correspondence, but the paper explicitly flags it as a conjecture: footnote 21 says 'this merely constitutes a well-motivated conjecture, as a full thimble decomposition ... is yet far out of reach', and the conclusion repeats that the full thimble decomposition 'remains an open problem'. An unproved conjecture is a missing proof, not a circular argument: the paper does not derive (4.8) from the spectral sum nor from a self-citation, and it does not fit the 1/2 factor to data. The only self-reference to [45] is a retrospective reinterpretation of excited-state results via (4.11) and is not load-bearing. The biorthogonal completeness used in (3.9) is also acknowledged to be non-rigorous in infinite dimensions, another limitation rather than a circularity. Hence no step reduces by construction to its own input; score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on spectral-theoretic assumptions about polynomial ODEs, on an unproved completeness assumption for the non-Hermitian contour Hamiltonian, on a conjectural thimble decomposition, and on a potential deformation whose nonperturbative effects are acknowledged to be ambiguous. There are no fitted numerical parameters.

assumptions (5)

standard math The generalized eigenvalue problem (2.7) with subdominance in two non-adjacent Stokes sectors has a discrete set of simple eigenvalues whose moduli accumulate at infinity.
This is a theorem from the exact WKB / Sibuya literature invoked in section 2.1; the paper relies on it to define the spectrum it later extracts.
ad hoc to paper The bi-orthogonal eigenfunctions ψℓ and ϕℓ of bHγ and bH†γ form a complete basis in L2(R) with resolution of unity (3.9), even though footnote 8 notes the rigorous statement is finite-dimensional.
The spectral representation (3.11) and the master formula (3.34) require this completeness, which is not proven for the non-Hermitian contour Hamiltonian.
ad hoc to paper The discretized path integral on any contour Γ satisfying the local constraint (C.1) and the global convergence condition (3.32) converges to the same continuum limit, and contours that violate (C.1) can be obtained by pinching.
The path integral derivation in section 3.2 and Appendix C requires this to hold, including for Stokes sectors that lie within one 90-degree sector.
ad hoc to paper The steepest-descent decomposition of the infinite-dimensional complexified function space for the resonant propagator is C([0,T], Γ) = J_FV + J_bounce with the overlap factor 1/2.
This is explicitly stated in section 4.3 to be a well-motivated conjecture; the paper's central application depends on it.
domain assumption The potential deformation V(stable) to V(unstable) past z_escape leaves the barrier and FV region invariant and yields the same semiclassical decay rate up to nonperturbative corrections, despite the two potentials not being simultaneously analytic.
Used in section 4.3 to justify replacing the bounded potential by an unstable one; the paper acknowledges this introduces an ambiguity in defining the decay rate nonperturbatively.

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Pith. "Pith review of Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states." pith.science (2026). https://pith.science/paper/KW7NTGP7

@misc{pith2026250723125,
  author       = {Pith},
  title        = {Pith review of: Path integral analysis of Schr\"odinger-type eigenvalue problems in the complex plane: Establishing the relation between instantons and resonant states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KW7NTGP7}},
  note         = {Machine review of arXiv:2507.23125}
}

abstract

Schr\"odinger-type eigenvalue problems are ubiquitous in theoretical physics, with quantum-mechanical applications typically confined to cases for which the eigenfunctions are required to be normalizable on the real axis. However, seeking the spectrum of resonant states for metastable potentials or comprehending $\mathcal{PT}$-symmetric scenarios requires the broader study of eigenvalue problems for which the boundary conditions are provided in specific angular sectors of the complex plane. We generalize the conventional path integral treatment to such nonstandard boundary value problems, allowing the extraction of spectral information using functional methods. We find that the arising functional integrals are naturally defined on a complexified integration contour, encapsulating the demanded sectorial boundary conditions of the associated eigenvalue problem. The attained results are applied to the analysis of resonant ground-state energies, through which we identify the previously elusive one-to-one correspondence between decay rates derived from real-time quantum tunneling dynamics and those obtained via the Euclidean instanton method.

Figures

Figures reproduced from arXiv: 2507.23125 by the authors.

**Figure 1.** (Left) Prototypical potential exhibiting decay from the false vacuum (FV) region toward the global minimum zTV, subsequently referred to as the true vacuum (TV). The shaded barrier region is assumed to be sufficiently high (and broad) in order for the decay dynamics to be slow compared to the oscillation periods in the FV and TV region, constituting the natural time scales in such a potential. (Right) Given an exemp… view at source ↗

**Figure 2.** (Left) Stokes graph for the polynomial Q(z) = √ 3 + i z 4 − [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Examples of different admissible eigenvalue problems admitting a purely discrete spectrum—the regions of subdominance S± specifying the boundary conditions have been shaded. (Left) A conventional eigenvalue problem that can be defined on the real axis, being fully entailed in the Stokes sectors where the eigenfunctions are mandated to be subdominant. In that case, one encounters the typical requirement of the eigenf… view at source ↗

Figures from the paper (13 more)

**Figure 4.** Figure 4: Process of analytic continuation of the eigenvalue problem (2.8) given the exemplary quartic potential V (φ) (z) = 6e iφz 4 + z 3 + 3z 2 from φ = 0 to φ = −π, rendering the initially stable potential unstable in the process. Whereas the boundary value problem associate…

**Figure 5.** Figure 5: Illustration of the behavior of an odd-degree polynomial, with the example shown being a cubic potential with real coefficients. Once more, in the spatial direction for which barrier penetration effects are relevant, the real axis resides at the boundary of two Stokes …

**Figure 6.** Figure 6: Exemplary choice of the complex contour Γ, parametrized through the smooth surjection γ(•) : R → Γ, asymptotically terminating in the two non-adjacent Stokes wedges S± chosen for the definition of the generic Schrödinger-type eigenvalue problem (3.1). Any such contour …

**Figure 7.** Figure 7: Schematic illustration of the potential-deformation approach by Callan & Coleman [17]. In case the false vacuum zFV is stabilized, as shown in the left panel, the sole critical trajectory starting and ending at the (then global) minimum zFV for T → ∞ is the trivial FV …

**Figure 8.** Figure 8: Illustration of the analytic continuation of the single-dimensional integral ZE(g) from g = 1 to g = −1 along arg(g) < 0. The saddle points z0,± as well as their associated steepest ascent (blue, K0,±) and descent (red, J0,±) contours are shown for four different value…

**Figure 9.** Figure 9: Depiction of the relevant (real) classical trajectories emerging when investigating the Euclidean propagator K (global) E [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

**Figure 10.** Figure 10: Schematic structure of the relevant steepest-descent thimbles in the complexified function space C C [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

**Figure 11.** Figure 11: Schematic procedure on how to compute the resonant spectrum in the metastable FV region of an otherwise stabilized potential V (stable)(z). One initially deforms the original potential past the classical escape point zescape to be unbounded from below, rendering the p…

**Figure 12.** Figure 12: Expected structure of the steepest-descent thimbles for the deformed, unstable potential V (unstable)(z). The emerging eigenvalue problem is defined in tilted Stokes wedges bordering the real axis in all asymptotic regions into which the particle can tunnel. Conseque…

**Figure 13.** Figure 13: Depiction of the local restriction on the contour Γ from demanding the intermediate momentum integrals (3.17) to be well-defined. The constraint (C.1) is equivalent to requiring the slope of the contour to always be contained in a tilted light cone. The situation is s…

**Figure 14.** Figure 14: Illustration of the arising situation when trying to connect the asymptotic angular regions ϑ± by a contour Γ respecting the local constraint (C.1). (Left) For [PITH_FULL_IMAGE:figures/full_fig_p058_14.png]

**Figure 15.** Figure 15: Exemplary case of the combined constraints inflicted by demanding the contour Γ to asymptotically end inside the Stokes sectors S± as well as Sk. (Left) Usual Stokes sector picture portraying the wedges Sk, with the eigenvalue problem defined in the two highlighted se…

**Figure 16.** Figure 16: Case in which a Stokes sector has non-vanishing overlap with two distinct sectors Sk, thus giving rise to disjoint sectors in which one end of Γ can be chosen to terminate, yielding seemingly inequivalent functional integrals. However, due to the underlying eigenvalue…

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