Pith. sign in

REVIEW 1 major objections 3 minor 37 references

For a particle in a soft quantum waveguide coupled to a massive scalar field, the curved guide's essential spectrum is exactly the half-line starting at the smaller of the straight-guide threshold and the one-boson creation threshold.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 05:16 UTC pith:XF5VCO4W

load-bearing objection A credible and new spectral analysis of a Nelson model on a soft waveguide; the main results are plausible but one imported inclusion in the HVZ theorem needs independent proof. the 1 major comments →

arxiv 2607.29526 v1 pith:XF5VCO4W submitted 2026-07-31 math-ph math.MP

Spectral Analysis of a Soft Quantum Waveguide Coupled to a Massive Quantum Scalar Field

Benjamin Alvarez , Hugo Gouttenegre This is my paper
classification math-ph math.MP MSC 81Q1081T10
keywords soft quantum waveguideparticle-field Hamiltonianmassive scalar fieldessential spectrumHVZ theoremionization thresholdbinding conditionfiber decomposition
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies a nonrelativistic particle moving in a planar soft quantum waveguide—an infinite strip-shaped potential well—linearly coupled to a massive scalar bosonic field. Its central claim is that the curved waveguide has a gapless essential spectrum whose bottom is the smaller of two thresholds: the ionization threshold shared with the straight waveguide, and the energy needed to add one boson. The paper also proves the straight guide's spectrum is purely essential and gapless, and gives a sufficient condition for bound states: if an effective one-dimensional Schrödinger operator built from a fiber ground state has negative energy, then discrete spectrum appears below the essential one. The overall method treats the curved guide as a localized geometric perturbation of the straight guide, combining fiber decomposition, localization, Persson's formula, and an HVZ-type threshold argument.

Core claim

The central discovery is the HVZ-type identity σ_ess(H_C) = [min(Σ, inf σ(H_C)+m), ∞), where H_C is the curved-waveguide Hamiltonian, m is the boson mass, and Σ is the ionization threshold—equal to the straight guide's threshold and to inf σ(H_S). The essential spectrum is therefore a closed half-line with no gaps, and its bottom is set by whichever is lower: the geometric threshold of the straight guide or the threshold for creating one boson. A separate result reduces the question of bound states in the curved guide to a one-dimensional criterion: with Ψ the ground state of the fiber Hamiltonian at a minimizing longitudinal momentum, the effective potential V_eff,Ψ(x1)=∫(V_C−V_S)(x1,x2)‖Ψ(

What carries the argument

The straight Hamiltonian is decomposed into longitudinal-momentum fibers H_S(η) by a unitary Fourier-type transformation ('fibering operator') that exploits translation invariance along the guide; each fiber acts on L²(R_{x2}) tensor the Fock space, with the total longitudinal momentum η as a parameter. The curved Hamiltonian is compared to this straight one through IMS localization and Persson's formula, using field-alignment unitaries on the straight branches; this yields the common ionization threshold Σ and the essential-spectrum lower bound. The binding condition rests on an effective one-dimensional operator −∂²_{x1}+V_eff,Ψ, built from the fiber ground state Ψ, which translates the ge

Load-bearing premise

The curved Hamiltonian H_C has a ground state; this is explicitly adopted as an assumption in Proposition 4.18 ('We therefore retain the existence of a ground state as an assumption') rather than proved, and it is needed for the inclusion [inf σ(H_C)+m, ∞) ⊂ σ_ess(H_C) that completes the HVZ theorem when the one-boson threshold is the smaller one.

What would settle it

Compute, for a simple compactly curved soft guide (e.g., a finite circular arc) with a massive scalar field, the Persson-limit bottom of σ_ess(H_C) and compare it with min(Σ, inf σ(H_C)+m); any mismatch in either direction refutes Theorem 4.19.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The spectra of both straight and curved waveguides coupled to the massive field are gapless half-lines, so the coupling does not open gaps in the essential spectrum.
  • Discrete spectrum of H_C exists exactly when the geometry pulls inf σ(H_C) below Σ, the same geometric criterion as for waveguides without a field.
  • The one-boson threshold inf σ(H_C)+m can dominate the bottom of the essential spectrum when the field coupling is strong relative to the geometry.
  • The effective potential V_eff,Ψ gives a computable, testable criterion for curvature-induced bound states in the coupled model.
  • The common ionization threshold Σ = inf σ(H_S) anchors the whole spectral picture, so straight-guide computations directly constrain curved-guide physics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the load-bearing point most worth probing is the assumed ground state of H_C used for the inclusion [inf σ(H_C)+m, ∞) ⊂ σ_ess(H_C); if that assumption fails for some coupling or geometry, the HVZ formula's min may still hold via another route, or may break—a natural target for a follow-up proof or counterexample.
  • Editorial extension: because the second threshold is E_C + m, taking m→0 would make the two thresholds merge and likely change the spectral picture; the massive-boson assumption is therefore not a technical convenience but a physically meaningful regulator whose removal requires new infrared estimates.
  • Editorial extension: the effective one-dimensional operator suggests a concrete numerical experiment—compute the straight-guide fiber ground state, build V_eff,Ψ for a compactly bent strip, and test for negative eigenvalues; a positive result would predict a curvature-induced bound state in the fully coupled model.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper studies a Nelson-type Hamiltonian describing a non-relativistic particle in a planar soft quantum waveguide linearly coupled to a massive scalar bosonic field. The authors prove self-adjointness and lower semiboundedness, establish Persson-type localization estimates, and analyze the straight waveguide by a longitudinal-momentum fiber decomposition to show that its spectrum is a gapless half-line (Theorem 4.14). For the curved waveguide, they prove an HVZ-type formula for the bottom of the essential spectrum (Theorem 4.19), and then derive a sufficient condition, expressed through the one-dimensional effective operator -∂²_x1 + V_eff,Ψ, under which the curved Hamiltonian has discrete spectrum below the essential one (Theorem 5.11). The main theorems are supported by extensive technical sections including IMS localization, exponential decay, extended Fock-space arguments, and a Q-space positivity argument for the fiber ground state.

Significance. If all technical steps are valid, this is a meaningful first rigorous spectral analysis of a soft quantum waveguide coupled to a quantized field. The paper does not fit the standard confined-particle or decaying-potential frameworks, so the adaptation of HVZ and binding-condition methods to a non-confining waveguide is a genuine contribution. Credit is due for the detailed self-adjointness and form-domain analysis, the Lee-Low-Pines fibering of the straight Hamiltonian, the positivity-preserving semigroup argument that yields η0 = 0, and the clean trial-state computation in Theorem 5.11. The central obstruction is Proposition 4.18, where the decisive half-line inclusion for σ_ess(H_C) is imported from [DG99] without a complete verification of its hypotheses in this non-compact model. Since Theorem 4.19 depends on that inclusion in the case E_C + m < Σ, the main spectral formula is not yet fully established as it stands.

major comments (1)
  1. [§4.3, Proposition 4.18 and Theorem 4.19] The proof of Theorem 4.19 in the case E_C + m < Σ relies on Proposition 4.18, which asserts σ_ess(H) ⊃ [inf σ(H) + m, ∞) by citing the ⊃-part of [DG99, Theorem 4.1]. The text explicitly states that the small system here does not have compact resolvent and that 'we therefore retain the existence of a ground state as an assumption', yet the paper does not verify the hypotheses (H1) and (I1) of [DG99] for the soft-waveguide Nelson Hamiltonian, nor does it provide a self-contained Weyl-sequence construction for the inclusion. The waveguide potential is non-confining, does not vanish at infinity, and the small system lacks compact resolvent, so the transfer from the confined-particle framework of [DG99] is not automatic. This is load-bearing: without this inclusion, the bottom of σ_ess(H_C) could be strictly larger than min(Σ, E_C + m), and the exact spectral formula in Theorem 4.19 would fai
minor comments (3)
  1. [§4.1, Proposition 4.4] The proof says that hypotheses (i) and (ii) of [Gri04, Theorem 1] 'are only used to establish hypothesis (iii) after closure'. This is too terse; please spell out how the closedness of q_H replaces those assumptions, otherwise the reader cannot check the applicability of the cited theorem.
  2. [§4.2, Lemma 4.7 and Theorem 4.9] The notation F_{x1} for the partial Fourier transform is introduced without a precise domain statement. Since the fibering operator acts on vector-valued functions, please state explicitly that F_{x1} is taken on L²(R_{x1}, L²(R_{x2}, F_s)) and indicate the chosen convention for the Fourier transform.
  3. [§5.2, Theorem 5.11] Minor typographical issue: in the display after the definition of V_eff,Ψ, the inequality should read ⟨u|(-∂²_x1 + V_eff,Ψ(X1))u⟩_{L²} + Σ, not with an extra factor; the same expression is written correctly in the following line. Please check consistency.

Circularity Check

0 steps flagged

No significant circularity: the central HVZ and binding-condition derivations are self-contained or rely on external theorems, not on fitted inputs or self-citation chains.

full rationale

The paper's derivation chain does not exhibit a circular reduction. The straight-waveguide spectral half-line (Theorem 4.14) is proved from the fiber decomposition, continuity and growth of the fiber energies, and a translation-invariance argument for the ionization threshold; none of these steps defines its conclusion into its hypothesis. The curved-waveguide HVZ theorem (Theorem 4.19) combines Proposition 4.5's lower bound (from external [FGS02] and [Gri04] results, with the paper's own exponential-decay verification) with Proposition 4.17's upper half-line (proved by strong resolvent convergence) and Proposition 4.18's imported inclusion from [DG99]. The one load-bearing transfer, Proposition 4.18, is explicitly flagged by the authors as an assumption: 'their argument for the existence of a ground state relies on the compact resolvent of the small-system, which is not available here. We therefore retain the existence of a ground state as an assumption.' This is an admitted limitation in verifying an external theorem's hypotheses, not a circular step: the ground state is a well-defined object whose existence in the relevant case E_C+m<Σ is supplied by Proposition 4.5, which is independent of the inclusion being imported. If the [DG99] transfer is invalid, the HVZ formula would be unsupported, but that is a correctness risk, not circularity. The binding condition (Theorem 5.11) is a genuine sufficient condition: it constructs a trial state from the fiber ground state Ψ and shows a negative-energy state of an effective one-dimensional operator implies discrete spectrum below Σ. This does not reduce to its own conclusion because Ψ and the effective operator are defined from the straight/fiber Hamiltonian, not from the unknown discrete spectrum of H_C. There are no fitted parameters called predictions, no self-citations are load-bearing (indeed the authors cite no prior work of their own), and no external 'uniqueness theorem' is invoked to force the model choice. The admitted non-compactness issue and the unverified hypotheses of [DG99] are honest limitations, but they do not make the derivation circular.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claim rests on a set of modeling hypotheses about the geometry, potential, and field coupling, plus standard theorems imported from the literature. No free parameters are fitted to data; no new entities are postulated. The most fragile ingredient is the asserted adaptation of [DG99, Thm 4.1] to the non-confined waveguide setting.

axioms (6)
  • domain assumption Geometric hypotheses (H1geom)-(H4geom): curve γ is C4, unit-speed, non-self-intersecting; curvature κ ∈ C2_0(R) compactly supported; strip does not meet the cut locus; curve is not U-shaped.
    Defines the soft waveguide and guarantees that at infinity the two branches decouple and the straightening map is a diffeomorphism; used in Persson-type localization (Prop 3.11, 4.15).
  • domain assumption Potential well V (H1pot): non-positive V ∈ L∞, V ≠ 0, supported in [-a,a].
    Defines the waveguide potential; ensures the transverse operator has a negative eigenvalue ε0 used in the binding proof.
  • domain assumption Massive scalar field (H1field): ω(k)=√(k²+m²), m>0.
    Massive bosons avoid infrared issues and give the threshold E_C+m and gapless essential spectrum.
  • domain assumption Form factor (H2field): v ∈ L²(R²_k), v_x(k)=v(k)e^{-ik·x}.
    Makes the Segal field operator well-defined and the interaction self-adjoint.
  • domain assumption Rotation invariance (H3field) and realness (H4field) of v, used in §4.3 and §5.1.
    H3field allows aligning the field with the two straight branches of the curved waveguide (Prop 4.15); H4field yields E(0) ≤ E(η) via positivity improving semigroups (Thm 5.9).
  • standard math External spectral theorems from [Gri04, Thm 1], [FGS02, Thm 2], [DG99, Thm 4.1], [Tak25], and [RS79] are accepted as stated or with the claimed adaptation.
    Used to prove exponential decay (Prop 4.4), lower bound on essential spectrum (Prop 4.5), spectral inclusion (Prop 4.18), and fiber binding estimate (Thm 5.2).

pith-pipeline@v1.3.0-daily-deepseek · 55883 in / 17726 out tokens · 150850 ms · 2026-08-03T05:16:05.463687+00:00 · methodology

0 comments
read the original abstract

We study a Nelson-type Hamiltonian describing a massive spinless non-relativistic particle moving in a planar soft quantum waveguide and linearly coupled to a massive scalar bosonic field. Our spectral analysis relies on comparing the curved waveguide with its straight counterpart. For the straight Hamiltonian, longitudinal translation invariance yields a fiber decomposition, and we prove that its spectrum is purely essential and gapless. For the curved Hamiltonian, we establish an HVZ-type formula for the bottom of the essential spectrum, involving that of the straight system and the one-boson threshold. We also derive a sufficient condition for the existence of non-empty discrete spectrum in the curved case, expressed in terms of an effective one-dimensional Schr\"odinger operator.

Figures

Figures reproduced from arXiv: 2607.29526 by Benjamin Alvarez, Hugo Gouttenegre.

Figure 1
Figure 1. Figure 1: Straight case: operator −∆ + VS(X). x1 x2 Perturbation, particle entrapped Spectrum: ×· · ·× Eel might be empty [ ε0 [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Curved case: operator −∆ + VC(X). 3. Preliminary results In this section, we establish some straightforward preliminary results in order to get a better understanding of the subject. 3.1. Self-adjointness, domain and associated quadratic form The first preliminary result is obviously the self-adjointness of H, for which we need the self￾adjointness and lower semi-boundedness of Hel, Hf . Lemma 3.1. The ope… view at source ↗
Figure 3
Figure 3. Figure 3: Spectra of the components of the free Hamiltonian, α = min(ε0, Eel+m). Remark 3.8. Note that this only holds because we consider massive bosons: otherwise, we would have σ [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The use of Persson’s formula for the quantum waveguide. We now turn this idea into a new proof of [Exn20, Proposition 3.1], as announced above. Proposition 3.11. The Hamiltonians hel,S and hel,C, corresponding to the straight and curved waveguides respectively, have the same essential spectrum. Proof. We rely on the Weyl-sequence construction from the proof of [Exn20, Proposition 3.1], which shows that eve… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic representation of the partition of unity used in the proof. The cutoffs ejR,± are supported in ΩR,±, whereas ejR, which is not shown in the figure, is supported outside the green regions. Only part of the cut locus is depicted for illustration, and the figure is not to scale. Using the IMS localization formula for Schrödinger operators (see, e.g., [CFKS08, Theorem 3.2]), one has, for ψ ∈ DR with … view at source ↗
Figure 6
Figure 6. Figure 6: Definition of the signed curvature κ of a curve γ. Up to Euclidean transformations, γ is uniquely determined by κ. Indeed, for all s0, s ∈ R, one sets β(s, s0) := R s s0 κ(t) dt and γ(s) = γ(s0) + R s s0 cos β(t, s0)dt, − R s s0 sin β(t, s0)dt . To parameterize the strip Ω a , we introduce the curvilinear coordinates (s, u) 7→ γ(s) + uN(s), where s is the arc length, u the normal distance, and N(s) = (−γ… view at source ↗
Figure 7
Figure 7. Figure 7: Local curvilinear coordinates. With these variables, the bent strip is defined as Ω a = {φ(s, u)| s ∈ R, u ∈] − a, a[ }. To prevent self-intersections, each point in Ω a must correspond to a unique pair (s, u), meaning φ must be a global diffeomorphism from R×] − a, a[ to Ω a . While the condition a∥κ∥∞ < 1 ensures φ is a local diffeomorphism (which can be verified using its Jacobian), global injectivity i… view at source ↗
Figure 8
Figure 8. Figure 8: Infinite strip Ω a of width 2a with a compactly supported curvature κ [PITH_FULL_IMAGE:figures/full_fig_p039_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the cut locus construction. The point xa is reached from the normal line issued from γ(s1), so xa = φ(s1, u1). The point xb is reached by two shortest normal lines, issued from γ(s1) and γ(s3); hence xb ∈ Cut(γ) and c+(s1) = c+(s3) = u2. For u > c+(s1), the point φ(s1, u) is no longer described by the half-normal line issued from γ(s1) in the one-to-one coordinate system; for instance, xc i… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 12 canonical work pages

  1. [1]

    a user Classics, Birkh \

    Werner O. Amrein, Anne Boutet de Monvel, and Vladimir Georgescu, C_0 -groups, commutator methods and spectral theory of N -body hamiltonians , Modern Birkh \"a user Classics, Birkh \"a user, Basel, 2013, Reprint of the 1996 edition; originally published as volume 135 in Progress in Mathematics, https://doi.org/10.1007/978-3-0348-0733-3 doi:10.1007/978-3-0...

    work page doi:10.1007/978-3-0348-0733-3 2013

  2. [2]

    10, 1121--1146, https://arxiv.org/abs/math-ph/0412069 arXiv:math-ph/0412069 , https://doi.org/10.1002/mma.718 doi:10.1002/mma.718

    Laurent Amour, Beno \^ t Gr \'e bert, and Jean-Claude Guillot, The dressed nonrelativistic electron in a magnetic field, Mathematical Methods in the Applied Sciences 29 (2006), no. 10, 1121--1146, https://arxiv.org/abs/math-ph/0412069 arXiv:math-ph/0412069 , https://doi.org/10.1002/mma.718 doi:10.1002/mma.718

    Pith/arXiv arXiv 2006

  3. [3]

    3, 275--299, https://arxiv.org/abs/1611.06161 arXiv:1611.06161 , https://doi.org/10.4064/sm8757-4-2017 doi:10.4064/sm8757-4-2017

    Wolfgang Arendt and Marcel Kreuter, Mapping theorems for Sobolev spaces of vector-valued functions , Studia Mathematica 240 (2018), no. 3, 275--299, https://arxiv.org/abs/1611.06161 arXiv:1611.06161 , https://doi.org/10.4064/sm8757-4-2017 doi:10.4064/sm8757-4-2017

    Pith/arXiv arXiv 2018

  4. [4]

    Volume I : Abstract linear theory , Monographs in Mathematics, vol

    Herbert Amann, Linear and quasilinear parabolic problems. Volume I : Abstract linear theory , Monographs in Mathematics, vol. 89, Birkh \"a user, Basel, 1995, https://doi.org/10.1007/978-3-0348-9221-6 doi:10.1007/978-3-0348-9221-6

    doi:10.1007/978-3-0348-9221-6 1995

  5. [5]

    Amrein, Hilbert space methods in quantum mechanics , 1st ed., Fundamental Sciences: Physics, EPFL Press, Lausanne, 2009, Distributed by CRC Press

    Werner O. Amrein, Hilbert space methods in quantum mechanics , 1st ed., Fundamental Sciences: Physics, EPFL Press, Lausanne, 2009, Distributed by CRC Press

    2009

  6. [6]

    Asao Arai, Analysis on Fock spaces and mathematical theory of quantum fields: An introduction to mathematical analysis of quantum fields , 2nd ed., World Scientific, Singapore, 2024, https://doi.org/10.1142/13735 doi:10.1142/13735

    work page doi:10.1142/13735 2024

  7. [7]

    E Low-Dimens

    Nzar Rauf Abdullah, Chi-Shung Tang, Andrei Manolescu, and Vidar Gudmundsson, Optical switching of electron transport in a waveguide- QED system , Phys. E Low-Dimens. Syst. Nanostruct. 84 (2016), 280--284, https://doi.org/10.1016/j.physe.2016.06.023 doi:10.1016/j.physe.2016.06.023

    work page doi:10.1016/j.physe.2016.06.023 2016

  8. [8]

    Translated by Carole Labrousse, https://doi.org/10.1002/9781118032725 doi:10.1002/9781118032725

    Jean-Pierre Aubin, Applied functional analysis, 2nd ed., Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs, and Tracts, John Wiley & Sons, Inc., New York, 2000, Exercises by Bernard Cornet and Jean-Michel Lasry. Translated by Carole Labrousse, https://doi.org/10.1002/9781118032725 doi:10.1002/9781118032725

    work page doi:10.1002/9781118032725 2000

  9. [9]

    u rg Fr \

    Volker Bach, J \"u rg Fr \"o hlich, and Israel Michael Sigal, Quantum electrodynamics of confined nonrelativistic particles, Advances in Mathematics 137 (1998), no. 2, 299--395, https://doi.org/10.1006/aima.1998.1734 doi:10.1006/aima.1998.1734

    arXiv 1998

  10. [10]

    Bordag, U

    M. Bordag, U. Mohideen, and V. M. Mostepanenko, New developments in the Casimir effect , Physics Reports 353 (2001), no. 1--3, 1--205, https://arxiv.org/abs/quant-ph/0106045 arXiv:quant-ph/0106045 , https://doi.org/10.1016/S0370-1573(01)00015-1 doi:10.1016/S0370-1573(01)00015-1

    Pith/arXiv arXiv 2001

  11. [11]

    Cycon, Richard G

    Hans L. Cycon, Richard G. Froese, Werner Kirsch, and Barry Simon, Schr \"o dinger operators: With applications to quantum mechanics and global geometry , Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, 2008, Corrected and extended second printing; original edition published in 1987 as a Springer Study Edition, https://doi.org/10.1007...

    doi:10.1007/978-3-540-77522-5 2008

  12. [12]

    108, Cambridge University Press, Cambridge, 1993

    Isaac Chavel, Riemannian geometry: A modern introduction , Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993

    1993

  13. [13]

    Selfadjointness , spectral theory and renormalisation , Ph.D

    Thomas Norman Dam, Non-perturbative results in quantum field theory. Selfadjointness , spectral theory and renormalisation , Ph.D. thesis, Aarhus University, 2018, See Paper D: Rigorous Results on the Bose--Polaron . https://data.math.au.dk/publications/phd/2018/math-phd-2018-tnd.pdf

    2018

  14. [14]

    Derezi \'n ski and C

    J. Derezi \'n ski and C. G \'e rard, Asymptotic completeness in quantum field theory: Massive Pauli--Fierz hamiltonians , Reviews in Mathematical Physics 11 (1999), no. 4, 383--450, URL: https://www.imo.universite-paris-saclay.fr/ christian.gerard/publis/pauli-f.pdf, https://doi.org/10.1142/S0129055X99000155 doi:10.1142/S0129055X99000155

    doi:10.1142/s0129055x99000155 1999

  15. [15]

    Pavel Exner and Hynek Kova r \'i k, Quantum waveguides, Theoretical and Mathematical Physics, Springer, Cham, 2015, https://doi.org/10.1007/978-3-319-18576-7 doi:10.1007/978-3-319-18576-7

    doi:10.1007/978-3-319-18576-7 2015

  16. [16]

    Pavel Exner, Spectral properties of soft quantum waveguides, Journal of Physics A: Mathematical and Theoretical 53 (2020), no. 35, 355302, See also the corrigendum: Journal of Physics A: Mathematical and Theoretical 54 (2021), 099501, doi:10.1088/1751-8121/abdbc7, https://arxiv.org/abs/2006.15071v3 arXiv:2006.15071v3 , https://doi.org/10.1088/1751-8121/ab...

    Pith/arXiv arXiv 2020

  17. [17]

    9, 2360003, https://doi.org/10.1142/S0129055X23600036 doi:10.1142/S0129055X23600036

    , Geometrically induced spectral properties of soft quantum waveguides and layers, Reviews in Mathematical Physics 36 (2024), no. 9, 2360003, https://doi.org/10.1142/S0129055X23600036 doi:10.1142/S0129055X23600036

    work page doi:10.1142/s0129055x23600036 2024

  18. [18]

    Fr \"o hlich, M

    J. Fr \"o hlich, M. Griesemer, and B. Schlein, Asymptotic completeness for Rayleigh scattering , Annales Henri Poincar \'e 3 (2002), no. 1, 107--170, https://doi.org/10.1007/s00023-002-8614-9 doi:10.1007/s00023-002-8614-9

    doi:10.1007/s00023-002-8614-9 2002

  19. [19]

    1, 29--78, https://doi.org/10.1007/s00220-004-1111-x doi:10.1007/s00220-004-1111-x

    Vladimir Georgescu, Christian G \'e rard, and Jacob Schach M ller, Spectral theory of massless Pauli--Fierz models , Communications in Mathematical Physics 249 (2004), no. 1, 29--78, https://doi.org/10.1007/s00220-004-1111-x doi:10.1007/s00220-004-1111-x

    doi:10.1007/s00220-004-1111-x 2004

  20. [20]

    Lieb, and Michael Loss, Ground states in non-relativistic quantum electrodynamics, Inventiones Mathematicae 145 (2001), no

    Marcel Griesemer, Elliott H. Lieb, and Michael Loss, Ground states in non-relativistic quantum electrodynamics, Inventiones Mathematicae 145 (2001), no. 3, 557--595, https://arxiv.org/abs/math-ph/0007014 arXiv:math-ph/0007014 , https://doi.org/10.1007/s002220100159 doi:10.1007/s002220100159

    Pith/arXiv arXiv 2001

  21. [21]

    Griesemer, Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics, Journal of Functional Analysis 210 (2004), no

    M. Griesemer, Exponential decay and ionization thresholds in non-relativistic quantum electrodynamics, Journal of Functional Analysis 210 (2004), no. 2, 321--340, https://arxiv.org/abs/math-ph/0206024 arXiv:math-ph/0206024 , https://doi.org/10.1016/j.jfa.2003.06.001 doi:10.1016/j.jfa.2003.06.001

    Pith/arXiv arXiv 2004

  22. [22]

    1, 52--109, https://doi.org/10.1016/0022-1236(72)90057-2 doi:10.1016/0022-1236(72)90057-2

    Leonard Gross, Existence and uniqueness of physical ground states, Journal of Functional Analysis 10 (1972), no. 1, 52--109, https://doi.org/10.1016/0022-1236(72)90057-2 doi:10.1016/0022-1236(72)90057-2

    work page doi:10.1016/0022-1236(72)90057-2 1972

  23. [23]

    35, Springer, Singapore, 2019, https://doi.org/10.1007/978-981-32-9305-2 doi:10.1007/978-981-32-9305-2

    Fumio Hiroshima, Ground states of quantum field models: Perturbation of embedded eigenvalues, SpringerBriefs in Mathematical Physics, vol. 35, Springer, Singapore, 2019, https://doi.org/10.1007/978-981-32-9305-2 doi:10.1007/978-981-32-9305-2

    work page doi:10.1007/978-981-32-9305-2 2019

  24. [24]

    2, De Gruyter Studies in Mathematics, no

    Fumio Hiroshima and J \'o zsef L o rinczi, Feynman--Kac -type theorems and Gibbs measures on path space: Applications in rigorous quantum field theory , 2nd ed., vol. 2, De Gruyter Studies in Mathematics, no. 34/2, De Gruyter, Berlin/Boston, 2020, https://doi.org/10.1515/9783110403541 doi:10.1515/9783110403541

    doi:10.1515/9783110403541 2020

  25. [25]

    Hislop and Israel M

    Peter D. Hislop and Israel M. Sigal, Introduction to spectral theory: With applications to Schr \"o dinger operators , 1st ed., Applied Mathematical Sciences, vol. 113, Springer-Verlag, New York, 1996, https://doi.org/10.1007/978-1-4612-0741-2 doi:10.1007/978-1-4612-0741-2

    work page doi:10.1007/978-1-4612-0741-2 1996

  26. [26]

    Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer, Berlin, Heidelberg, 1995, Reprint of the 1980 edition, https://doi.org/10.1007/978-3-642-66282-9 doi:10.1007/978-3-642-66282-9

    doi:10.1007/978-3-642-66282-9 1995

  27. [27]

    30, 30LT01, https://doi.org/10.1088/1751-8121/abf05e doi:10.1088/1751-8121/abf05e

    Sylwia Kondej, David Krej c i r \'i k, and Jan K r \'i z , Soft quantum waveguides with an explicit cut-locus, Journal of Physics A: Mathematical and Theoretical 54 (2021), no. 30, 30LT01, https://doi.org/10.1088/1751-8121/abf05e doi:10.1088/1751-8121/abf05e

    work page doi:10.1088/1751-8121/abf05e 2021

  28. [28]

    4, 883--889, https://doi.org/10.1103/PhysRev.92.883 doi:10.1103/PhysRev.92.883

    Tsung-Dao Lee and David Pines, Interaction of a nonrelativistic particle with a scalar field with application to slow electrons in polar crystals, Physical Review 92 (1953), no. 4, 883--889, https://doi.org/10.1103/PhysRev.92.883 doi:10.1103/PhysRev.92.883

    work page doi:10.1103/physrev.92.883 1953

  29. [29]

    The bottom of the spectrum , Annales Henri Poincar \'e 6 (2005), no

    Jacob Schach M ller, The translation invariant massive Nelson model: I . The bottom of the spectrum , Annales Henri Poincar \'e 6 (2005), no. 6, 1091--1135, https://doi.org/10.1007/s00023-005-0234-8 doi:10.1007/s00023-005-0234-8

    doi:10.1007/s00023-005-0234-8 2005

  30. [30]

    3, 305--330, https://arxiv.org/abs/math-ph/0609065 arXiv:math-ph/0609065 , https://doi.org/10.1016/S0034-4877(09)00014-7 doi:10.1016/S0034-4877(09)00014-7

    Annalisa Panati, Existence and nonexistence of a ground state for the massless Nelson model under binding condition , Reports on Mathematical Physics 63 (2009), no. 3, 305--330, https://arxiv.org/abs/math-ph/0609065 arXiv:math-ph/0609065 , https://doi.org/10.1016/S0034-4877(09)00014-7 doi:10.1016/S0034-4877(09)00014-7

    Pith/arXiv arXiv 2009

  31. [31]

    Arne Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schr \"o dinger operator , Mathematica Scandinavica 8 (1960), 143--153, https://doi.org/10.7146/math.scand.a-10602 doi:10.7146/math.scand.a-10602

    doi:10.7146/math.scand.a-10602 1960

  32. [32]

    Michael Reed and Barry Simon, Methods of modern mathematical physics, Academic Press, New York, 1972--1979, Volumes I--IV

    1972

  33. [33]

    265, Springer, Dordrecht, 2012, https://doi.org/10.1007/978-94-007-4753-1 doi:10.1007/978-94-007-4753-1

    Konrad Schm \"u dgen, Unbounded self-adjoint operators on Hilbert space , Graduate Texts in Mathematics, vol. 265, Springer, Dordrecht, 2012, https://doi.org/10.1007/978-94-007-4753-1 doi:10.1007/978-94-007-4753-1

    doi:10.1007/978-94-007-4753-1 2012

  34. [34]

    Herbert Spohn, Dynamics of charged particles and their radiation field, Cambridge University Press, Cambridge, 2004, https://doi.org/10.1017/CBO9780511535178 doi:10.1017/CBO9780511535178

    work page doi:10.1017/cbo9780511535178 2004

  35. [35]

    Toshimitsu Takaesu, A remark on the binding condition by the decay of particle's potential, 2025, Preprint, revised 25 November 2025, https://arxiv.org/abs/2409.14080v2 arXiv:2409.14080v2 , https://doi.org/10.48550/arXiv.2409.14080 doi:10.48550/arXiv.2409.14080

    work page doi:10.48550/arxiv.2409.14080 2025

  36. [36]

    Jan-Hendrik Treude, Ricci curvature comparison in Riemannian and Lorentzian geometry , Diplomarbeit, Albert-Ludwigs-Universit \"a t Freiburg, April 2011, URL: https://d-nb.info/1123465274/34

    arXiv 2011

  37. [37]

    3, 1435--1471, https://doi.org/10.1007/s00220-014-2273-9 doi:10.1007/s00220-014-2273-9

    Xue Ping Wang, Large-time asymptotics of solutions to the Kramers--Fokker--Planck equation with a short-range potential , Communications in Mathematical Physics 336 (2015), no. 3, 1435--1471, https://doi.org/10.1007/s00220-014-2273-9 doi:10.1007/s00220-014-2273-9

    doi:10.1007/s00220-014-2273-9 2015