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arxiv: 2505.16913 · v3 · submitted 2025-05-22 · 🧮 math-ph · math.MP· math.SP· nlin.CD· quant-ph

Quantum Systems with jump-discontinuous mass. I

Fabio Deelan Cunden , Giovanni Gramegna , Marilena Ligab\`o This is my paper

Pith reviewed 2026-05-22 01:37 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.SPnlin.CDquant-ph
keywords quantum mechanicsdiscontinuous masssemiclassical limitsspectral asymptoticsboundary conditionsself-adjoint operatorsone-dimensional systemstwo-torus
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The pith

A quantum particle with jump-discontinuous mass supports infinitely many distinct semiclassical limits, each labeled by a point on a spectral curve in the two-torus.

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

The paper studies a free quantum particle moving in one dimension where the mass jumps abruptly at isolated points. The Hamiltonian is realized as a self-adjoint version of the kinetic-energy operator, with the precise realization fixed by boundary conditions imposed at each mass discontinuity. For a particular scale-free family of these conditions, the eigenfunctions turn out to depend on energy in a highly sensitive and irregular manner. This sensitivity produces infinitely many different semiclassical limits, each corresponding to a distinct point on a spectral curve that sits inside the two-torus. The result shows how discontinuous coefficients together with boundary data can generate unexpectedly rich spectral asymptotics.

Core claim

For the kinetic-energy operator on the line with jump-discontinuous mass and scale-free boundary conditions at the jumps, the associated spectral problem yields eigenfunctions whose energy dependence is highly sensitive and erratic. Consequently the system admits infinitely many distinct semiclassical limits, each labeled by a point on a spectral curve embedded in the two-torus.

What carries the argument

The spectral curve embedded in the two-torus that parametrizes the distinct semiclassical limits arising from the erratic energy dependence of the eigenfunctions.

If this is right

  • The spectrum and eigenfunctions cannot be captured by a single semiclassical approximation.
  • Spectral asymptotics must be described by a curve-worth of limiting regimes rather than one.
  • The interplay between discontinuous mass and boundary data controls the structure of high-energy states.
  • Self-adjoint extensions are selected by the scale-free conditions to produce the observed erratic behavior.

Where Pith is reading between the lines

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

  • Similar multiple-limit phenomena may occur in other one-dimensional Schrödinger operators whose coefficients change abruptly.
  • Phase-space analysis on the two-torus could become a standard tool for asymptotics of variable-mass or piecewise-constant-coefficient systems.
  • Engineered mass profiles in ultracold-atom or semiconductor experiments might exhibit signatures of these distinct semiclassical regimes.
  • The construction suggests a general mechanism by which boundary data at discontinuities can enrich the semiclassical picture beyond the usual single-limit case.

Load-bearing premise

The boundary conditions at each mass discontinuity are taken from a specific scale-free family that fixes a unique self-adjoint realization of the kinetic-energy operator.

What would settle it

An explicit computation or numerical diagonalization that yields only a single or finitely many semiclassical limits for the chosen scale-free boundary conditions would disprove the claim of infinitely many limits.

Figures

Figures reproduced from arXiv: 2505.16913 by Fabio Deelan Cunden, Giovanni Gramegna, Marilena Ligab\`o.

Figure 2
Figure 2. Figure 2: Yet, amid this irregularity, order emerges in the form of upper limit, Ces [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Particle density |𝜓𝐸𝜅 | 2 for the first 20 values of 𝐸𝜅 = ℏ2 𝜅 2 /2. Here 𝑚1 = 16, 𝑚2 = 1, ℓ1 = e, and ℓ2 = 𝜋. The corresponding frequencies 𝜔1 = 4e and 𝜔2 = 𝜋 are nonresonant. In the plots we also indicated the corresponding values of 𝜅 and L (𝜓𝐸𝜅 ). The present work originates as an attempt to detect whether such phenomena persist in continuous, rather than lattice, systems. The idea was that in such a i… view at source ↗
Figure 2
Figure 2. Figure 2: Leaning L (𝜓𝐸𝜅 ) versus 𝜅. We also display the upper limit (1.10) (orange line), the Cesaro limit (1.11) (blue line), and the ` lower limit (1.12) (red line). Here the frequencies are nonresonant (same mass and length values as in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pictorial representation of the boundary conditions investi￾gated: (a) the segment, equation (4.10), (b) the ring, equation (4.25), (c) the pendant, equation (4.41), and (d) the rose, equation (4.55) To prove this we will see that for all 𝑗 = 1, . . . , 𝑞, the angle 𝜃 𝑗 between the normal to the set Σ𝑃, 𝑗 and the linear flow on the torus (3.9) is such that (4.9) cos 𝜃 𝑗 = 𝜔 · ∇ 𝑓𝑃, 𝑗(𝜑1, 𝜑2) ∥𝜔∥ ∥∇ 𝑓𝑃, 𝑗(𝜑… view at source ↗
Figure 4
Figure 4. Figure 4: Zero set Σ𝑃 of the spectral functions on the torus T 2 , along with the linear flow (3.9). Here 𝑚1 = 16, 𝑚2 = 1, ℓ1 = e, and ℓ2 = 2𝜋. The corresponding frequencies 𝜔1 = 4e and 𝜔2 = 𝜋 are nonresonant. Since 0 is not an eigenvalue of 𝐻I−2𝑃, the spectrum of 𝐻I−2𝑃 is (4.12) 𝜎(𝐻I−2𝑃) =  𝐸𝜅 = ℏ 2 2 𝜅 2 : 𝜅 ∈ K , where (4.13) K = {𝜅 ≠ 0 : 𝑓𝑃 (𝜔1𝜅, 𝜔2𝜅) = 0} . For all 𝜅 ∈ K the null space of the spectral matrix … view at source ↗
Figure 5
Figure 5. Figure 5: Barra-Gaspard measures 𝑑𝜈𝑃 (𝜑1) from (4.22)-(4.38)-(4.53)- (4.67) (see [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical values (black dots) of the leaning L (𝜓𝐸𝜅 ) versus 𝜅. The upper (in orange) and the lower (in red) limits of the band are the one computed in Sections 4.1- 4.2- 4.3- 4.4. The Cesaro limit in blue is ` also displayed. The parameters are the same of [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Integration regions (5.5) for the computation of 𝑊𝜓. Here, the integration regions in the variable 𝑦 (see [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Wigner function 𝑊𝜓𝐸𝜅 for an eigenfunction of 𝐻𝑈, see (5.7). Here 𝑈 corresponds to Kirchhoff b.c. at the junction 𝑥 = 0 and Dirichlet b.c. at the pendant vertices, and 𝜅 corresponds to the 74th energy level of 𝐻I−2𝑃. The parameters are the same of [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
read the original abstract

We consider a free quantum particle in one dimension whose mass profile exhibits jump discontinuities. The corresponding Hamiltonian is a self-adjoint realisation of the kinetic-energy operator, with the specific realisation determined by the boundary conditions at the points of mass discontinuity. For a family of scale-free boundary conditions, we analyse the associated spectral problem. We find that the eigenfunctions exhibit a highly sensitive and erratic dependence on the energy. Notably, the system supports infinitely many distinct semiclassical limits, each labeled by a point on a spectral curve embedded in the two-torus. These results demonstrate a rich interplay between discontinuous coefficients, boundary data, and spectral asymptotics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies a one-dimensional free quantum particle whose mass profile has jump discontinuities. The Hamiltonian is realized as a self-adjoint extension of the kinetic-energy operator, with the extension fixed by a family of scale-free boundary conditions imposed at each discontinuity. The central claim is that the associated spectral problem yields eigenfunctions with highly sensitive, erratic energy dependence, and that the system admits infinitely many distinct semiclassical limits, each parametrized by a point on a spectral curve embedded in the two-torus.

Significance. If the construction is correct, the result exhibits a concrete mechanism by which discontinuous coefficients together with scale-free boundary data can generate a positive-dimensional family of distinct high-energy asymptotics. This provides a mathematically explicit example of multiplicity in semiclassical limits that is not reducible to a single effective potential or a finite set of phases, and may serve as a benchmark for spectral theory of operators with singular or piecewise-constant coefficients.

major comments (2)
  1. [Abstract / Hamiltonian definition] Abstract and the paragraph defining the Hamiltonian: the claim that the chosen scale-free boundary conditions produce a well-defined spectral curve in T² with the property that distinct points generate genuinely different semiclassical limits is load-bearing. The manuscript must show explicitly that the matching conditions at each mass jump remain energy-independent in the E→∞ regime and that the resulting transfer matrix yields an embedded curve whose points correspond to distinct limiting distributions of eigenfunctions or eigenvalue spacings; otherwise the asserted infinitude of distinct limits collapses.
  2. [Spectral curve construction] The section constructing the spectral curve: it is not yet clear whether the high-energy phase accumulation across jumps is controlled uniformly for all points on the putative curve. If the erratic energy dependence noted in the abstract arises from rapid oscillations that average out differently for different boundary parameters, the limiting set may be a single point or a lower-dimensional subset rather than a curve of positive dimension; a concrete test (e.g., explicit computation of the limiting WKB phase or numerical sampling of high eigenvalues for two distinct curve points) is required.
minor comments (1)
  1. The abstract would benefit from a one-sentence statement of the explicit form of the mass profile (e.g., number and locations of jumps) to orient the reader before the spectral claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and the constructive major comments, which help clarify the presentation of our results on the spectral curve and its implications for distinct semiclassical limits. We address each point in turn and indicate the revisions planned for the next version of the manuscript.

read point-by-point responses

  1. Referee: [Abstract / Hamiltonian definition] Abstract and the paragraph defining the Hamiltonian: the claim that the chosen scale-free boundary conditions produce a well-defined spectral curve in T² with the property that distinct points generate genuinely different semiclassical limits is load-bearing. The manuscript must show explicitly that the matching conditions at each mass jump remain energy-independent in the E→∞ regime and that the resulting transfer matrix yields an embedded curve whose points correspond to distinct limiting distributions of eigenfunctions or eigenvalue spacings; otherwise the asserted infinitude of distinct limits collapses.

    Authors: The scale-free boundary conditions are introduced in Section 2 and are formulated without any dependence on the spectral parameter E; they involve only the values and derivatives of the wave function at the discontinuity points, scaled by the local mass values. The transfer matrix across each jump is therefore energy-independent by construction. In Section 3 we compose these matrices with the free propagation phases on each interval and extract the high-energy asymptotics, showing that the limiting phase map on the two-torus is parametrized by the boundary data. Distinct points on the resulting spectral curve produce distinct limiting rotation numbers and hence distinct distributions of eigenfunction oscillations. To make the argument fully explicit we will add a short subsection that writes the limiting transfer matrix in closed form and verifies that the associated invariant measures differ for generic pairs of curve points. revision: yes

  2. Referee: [Spectral curve construction] The section constructing the spectral curve: it is not yet clear whether the high-energy phase accumulation across jumps is controlled uniformly for all points on the putative curve. If the erratic energy dependence noted in the abstract arises from rapid oscillations that average out differently for different boundary parameters, the limiting set may be a single point or a lower-dimensional subset rather than a curve of positive dimension; a concrete test (e.g., explicit computation of the limiting WKB phase or numerical sampling of high eigenvalues for two distinct curve points) is required.

    Authors: We agree that uniform control of the phase accumulation must be stated more clearly. The total semiclassical phase is the sum of the free WKB integrals on each mass interval plus the fixed phase shifts from the scale-free conditions; because the mass is piecewise constant the integrals are linear in sqrt(E) and the boundary phases remain bounded independently of E. Consequently the limiting map is a continuous function of the boundary parameters and traces a curve of positive dimension in the torus for the family we consider. To supply the requested concrete test we will include, in the revised manuscript, an explicit evaluation of the limiting WKB phase for two representative points on the curve together with a brief numerical check of the high-energy eigenvalue spacings for those same parameters. revision: yes

Circularity Check

0 steps flagged

No circularity: spectral analysis proceeds from chosen boundary conditions without self-referential reduction

full rationale

The paper defines the Hamiltonian via a specific family of scale-free boundary conditions at mass discontinuities, then directly analyzes the resulting spectral problem to derive the erratic energy dependence of eigenfunctions and the existence of an embedded spectral curve in the two-torus parametrizing distinct semiclassical limits. This is a standard forward derivation from the operator domain and matching conditions; no step equates a derived quantity (such as the curve or the limits) to a fitted parameter or prior self-citation by construction. The abstract and reader's summary confirm the central claim rests on explicit analysis of the chosen self-adjoint realization rather than any load-bearing loop.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard functional-analysis axioms for self-adjoint extensions and on the modeling choice of scale-free boundary conditions; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The kinetic-energy operator admits self-adjoint realizations determined by boundary conditions at mass discontinuities.
    Invoked in the abstract when defining the Hamiltonian as a specific self-adjoint realization.

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