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arxiv: 2604.23011 · v2 · pith:4STDK4BMnew · submitted 2026-04-24 · 🧮 math-ph · math.MP

Effect of kinetic energy operator on double heterostructures spectra

R. Valencia-Torres , J. Garc\'ia-Ravelo , E. Chore\~no-Ortiz , J. Avenda\~no This is my paper

Pith reviewed 2026-05-25 06:56 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords kinetic energy operatordouble heterostructuresposition-dependent massenergy spectrareflection coefficienttranscendental equationbound states
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The pith

Different kinetic energy operators produce distinct bound-state spectra in double heterostructures with position-dependent mass.

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

The paper calculates the energy spectra of double heterostructures with position-dependent mass using two methods for multiple kinetic energy operators. The transcendental equation is used when analytically feasible, while the poles of the reflection coefficient provide spectra for any such structure. These results complement previous spectra by adding another operator and offer a means to critically examine claims about operator relevance in simpler heterostructures. This matters because different operators can lead to different quantum energy levels in systems where mass is not constant.

Core claim

Using two complementary techniques—the transcendental equation method for feasible cases and the poles of the reflection coefficient for arbitrary double heterostructures—the energy spectra are calculated for various kinetic energy operators. These spectra complement those from prior references by including another operator and enable critical analysis of statements about the relevance of certain operators in simpler heterostructures.

What carries the argument

The poles of the reflection coefficient of the heterostructure, which locate the bound-state energies for arbitrary double heterostructures, together with the transcendental equation obtained from an analytical process for specific cases.

If this is right

  • The spectra for an additional kinetic energy operator complement those already reported in previous references.
  • These methods allow critical analysis of statements that have been made about the relevance of some kinetic energy operators in some simpler heterostructures.
  • The reflection coefficient approach can be used practically in any double heterostructure.

Where Pith is reading between the lines

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

  • If the reflection coefficient method is reliable, it could be applied to test consistency of operator choices in triple heterostructures or other extended potential profiles.
  • Experimental spectra from position-dependent mass systems could be compared against these predictions to assess which operator better matches physical observations.

Load-bearing premise

That the poles of the reflection coefficient correctly yield the bound-state energy spectra for arbitrary double heterostructures, and that the transcendental equation method applies without additional unstated conditions when feasible.

What would settle it

A direct numerical solution of the position-dependent mass Schrödinger equation for a chosen double heterostructure and kinetic energy operator whose bound-state energies fail to match the reflection coefficient poles or the roots of the transcendental equation.

Figures

Figures reproduced from arXiv: 2604.23011 by E. Chore\~no-Ortiz, J. Avenda\~no, J. Garc\'ia-Ravelo, R. Valencia-Torres.

Figure 1
Figure 1. Figure 1: Potential V (z) (gray line) of DH (42) and its eigenfunctions; ψ 1 in(z) and ψ 2 in(z) are given by Eq. (48). The energy spectrum {E0, E1, E2, E3} = {4.63268, 10.0389, 15.223, 20.076} how was obtained using Eq. (22) correspond to the Z-K conditions accordingly with view at source ↗
Figure 2
Figure 2. Figure 2: a) First three eigenfunctions (17) of the system (49), view at source ↗
Figure 3
Figure 3. Figure 3: Wave function (17) with ψ 1 in(z), ψ 1 in(z) given by Eqs. (A.4) for the distributions (A.1). The energy val￾ues, {E0, E1, E2, E3, E4, E5, E6} = {−8.25, −6.875, −5.625, −4.50009, −3.5013, −2.63724, −1.96428}, are found through the trascendental equation (22) using the BD-D condition. E0 E1 E2 V(z) -1.0 -0.5 0.0 0.5 1.0 z -9 -8 -7 -6 -5 -4 -3 view at source ↗
Figure 4
Figure 4. Figure 4: Wave function (17) with ψ 1 in(z), ψ 1 in(z) given by Eqs. (A.9). The energy values {E0, E1, E2} are found in view at source ↗
read the original abstract

We report on the effect of the kinetic energy operator ambiguity on the energy spectra of various double heterostructures when the mass of the charge carriers, subjected to a potential, depends on position. The spectra are calculated using two complementary techniques. In the first case, which is not always possible, the energy spectrum satisfies a transcendental equation, which is obtained through an analytical process. While in the second, which can be used practically in any double heterostructure (DH), the energy spectra correspond to the poles of the reflection coefficient of the heterostructure. The spectra thus obtained complement those already reported in a previous reference, since we now include the spectra of another kinetic energy operator (KEO). Finally, we show how these methods allow us to critically analyze some statements that have been made about the relevance of some kinetic energy operators in some simpler heterostructures.

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

1 major / 1 minor

Summary. The manuscript examines the effect of different kinetic energy operators on the energy spectra of double heterostructures with position-dependent carrier mass. Spectra are obtained via two complementary techniques: an analytical transcendental equation (when feasible) and the poles of the reflection coefficient (presented as generally applicable). The work extends prior results by including spectra for an additional KEO and uses the methods to critically analyze statements on KEO relevance in simpler heterostructures.

Significance. If the reflection-coefficient poles are rigorously shown to correspond to bound states for arbitrary KEOs, the results would usefully complement existing literature on position-dependent-mass problems and provide a basis for evaluating KEO choices. The absence of explicit derivations, error analysis, or validation against known limits in the provided description reduces the assessed significance of the central claims.

major comments (1)
  1. [Abstract] Abstract: the assertion that energy spectra 'correspond to the poles of the reflection coefficient of the heterostructure' for any double heterostructure and any KEO lacks an explicit derivation. Different KEO orderings (BenDaniel-Duke, Zhu-Kroemer, etc.) alter the form of the Schrödinger equation, the interface matching conditions, and the definition of the asymptotic current; without showing that the pole-to-bound-state mapping survives these changes, the complementary spectra and the subsequent critical analysis rest on an unverified assumption.
minor comments (1)
  1. The qualifier 'when feasible' for the transcendental-equation method is left undefined; specifying the precise conditions (e.g., symmetry requirements or potential forms) under which the method applies would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the detailed comment. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that energy spectra 'correspond to the poles of the reflection coefficient of the heterostructure' for any double heterostructure and any KEO lacks an explicit derivation. Different KEO orderings (BenDaniel-Duke, Zhu-Kroemer, etc.) alter the form of the Schrödinger equation, the interface matching conditions, and the definition of the asymptotic current; without showing that the pole-to-bound-state mapping survives these changes, the complementary spectra and the subsequent critical analysis rest on an unverified assumption.

    Authors: We agree that the abstract asserts general applicability of the reflection-coefficient pole method without an explicit derivation of the pole-to-bound-state correspondence. In the body of the manuscript the reflection coefficient is constructed case-by-case using the probability current and interface conditions appropriate to each KEO (including the additional operator treated here), and the resulting poles are shown to reproduce the spectra obtained from the transcendental equation when the latter exists. Nevertheless, a single general derivation that the correspondence survives arbitrary changes in the differential operator, matching rules, and current definition is not supplied. We will add this derivation (most likely as an appendix) in the revised manuscript so that the claim for arbitrary KEOs and the subsequent critical analysis rest on a verified foundation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard QM methods

full rationale

The paper computes spectra for double heterostructures via two standard techniques: (1) transcendental equations obtained analytically from the position-dependent-mass Schrödinger equation under different KEO orderings, and (2) poles of the reflection coefficient. These are presented as complementary, with the transcendental route used “when feasible” and the reflection-coefficient route as the general tool. No equations or claims reduce a derived quantity to a fitted input by construction, nor does any load-bearing step rest solely on a self-citation whose content is unverified. The mention of a “previous reference” supplies complementary spectra for one additional KEO but does not justify the mapping from poles to bound states or the critical analysis; both follow from the explicit application of the two methods to the structures. The derivation chain is therefore self-contained against external benchmarks of quantum mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated. Standard quantum-mechanical assumptions (Schrödinger equation, position-dependent mass) are implicit but not detailed.

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