Effect of kinetic energy operator on double heterostructures spectra

E. Chore\~no-Ortiz; J. Avenda\~no; J. Garc\'ia-Ravelo; R. Valencia-Torres

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: 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: a) First three eigenfunctions (17) of the system (49), view at source ↗

**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 values, {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: 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.

Desk Editor's Note private letter to a colleague

Adds spectra for one more KEO using reflection poles but the mapping to bound states across orderings is not shown.

read the letter

The main point is that this paper calculates energy spectra for double heterostructures with position-dependent mass for one additional kinetic energy operator, using transcendental equations where they work and poles of the reflection coefficient as the general route. It complements an earlier reference and uses the results to revisit some prior claims about which operators matter in simpler cases. The calculations follow standard analytical and scattering methods for these systems, which is straightforward and produces concrete spectra for comparison. That part is useful for anyone already running similar models. The soft spot is the unshown step that the poles of the reflection coefficient still locate the bound states once the kinetic energy operator changes. Different orderings alter the form of the Schrödinger equation and the interface matching conditions, so the definition of the reflection coefficient itself shifts. The abstract flags that the transcendental route only applies when feasible, leaving the reflection method as the fallback, yet supplies no derivation or limit check confirming the poles remain valid. If that link fails for even one ordering, the new spectra and the critical comments both rest on weaker ground. This is narrow technical work aimed at specialists who already care about position-dependent mass in heterostructures. A reader who knows the previous paper might pull the extra numbers for direct comparison, but it does not introduce new methods or resolve the ordering ambiguity at a deeper level. Send it to a referee who can check the derivations and the reflection-coefficient justification rather than desk-rejecting outright.

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)

[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)

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

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.

pith-pipeline@v0.9.0 · 5693 in / 1022 out tokens · 20719 ms · 2026-05-25T06:56:49.066109+00:00 · methodology

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discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

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Mazharimousavi

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J. F. Cari˜ nena, A. M. Perelomov, M. F. Ra˜ nada, and M. Santander. A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator.Journal of Physics A: Mathematical and Theoretical, 41(8), 2008

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S. M. Ikhdair and R. Sever. Relativistic and nonrelativistic bound states of the isotonic oscillator by nikiforov- uvarov method.Journal of Mathematical Physics, 52(12), 2011

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J. Sesma. The generalized quantum isotonic oscillator.Journal of Physics A: Mathematical and Theoretical, 43(18), 2010

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V. V. Dodonov, V. I. Man’ko, and L. Rosa. Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential.Phys. Rev. A, 57:2851–2858, Apr 1998

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Dodonov, I.A

V.V. Dodonov, I.A. Malkin, and V.I. Man’ko. Even and odd coherent states and excitations of a singular oscillator.Physica, 72(3):597–615, 1974

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S. M. Nagiyev, E. I. Jafarov, and R. M. Imanov. On a dynamical symmetry group of the relativistic linear singular oscillator.Europhysics Letters, 76(2):175, sep 2006

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Self-adjoint extension procedure for a singular oscillator

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Abramowitz and I.A

M. Abramowitz and I.A. Stegun.Handbook of Mathematical Functions: With Formulas, Graphs, and Mathe- matical Tables. Applied mathematics series. Dover Publications, 1965. 16 6 Tables Symmetric Double Heterostructure (2) V(z) m(z) -4 -2 0 2 4 z -5 0 5 10 15 Vin(z) =−µ 2/(1 +z 2), m in(z) =σ 2/(1 +z 2) V0 =V 2 =V in(z0),m 0 =m 2 =m in(z0). σ= 4,µ= 3,z 0 =−z ...

work page 1965

[1] [1]

von Roos

O. von Roos. Position-dependent effective masses in semiconductor theory.Phys. Rev. B, 27, 1983

work page 1983

[2] [2]

Thomsen, G

J. Thomsen, G. T. Einevoll, and P. C. Hemmer. Operator ordering in effective-mass theory.Phys. Rev. B, 39, 1989

work page 1989

[3] [3]

de Souza Dutra and C.A.S

A. de Souza Dutra and C.A.S. Almeida. Exact solvability of potentials with spatially dependent effective masses.Physics Letters A, 275(1), 2000

work page 2000

[4] [4]

Valencia, J

R. Valencia, J. Avenda˜ no, A. Bernal, and J. Garc´ ıa. Energy spectra of position-dependent masses in double heterostructures.Physica Scripta, 95(7), 2020

work page 2020

[5] [5]

R. A. Morrow and K. R. Brownstein. Model effective-mass hamiltonians for abrupt heterojunctions and the associated wave-function-matching conditions.Phys. Rev. B, 30, 1984

work page 1984

[6] [6]

R. A. Morrow. Establishment of an effective-mass hamiltonian for abrupt heterojunctions.Phys. Rev. B, 35, 1987

work page 1987

[7] [7]

G. T. Einevoll and P. C. Hemmer. The effective-mass hamiltonian for abrupt heterostructures.Journal of Physics C: Solid State Physics, 21(36), 1988

work page 1988

[8] [8]

Hagston, P

W.E. Hagston, P. Harrison, T. Piorek, and T. Stirner. Boundary conditions on current carrying states and the implications for observations of bloch oscillations.Superlattices and Microstructures, 15(2), 1994

work page 1994

[9] [9]

Balian, D

R. Balian, D. Bessis, and G. A. Mezincescu. Form of kinetic energy in effective-mass hamiltonians for het- erostructures.Phys. Rev. B, 51, 1995

work page 1995

[10] [10]

M. J. Godfrey and A. M. Malik. Boundary conditions and spurious solutions in envelope-function theory.Phys. Rev. B, 53, 1996

work page 1996

[11] [11]

Balian, D

R. Balian, D. Bessis, and G.A. Mezincescu. Mesoscopic description of heterojunctions.J. Phys. I France, 6(10), 1996

work page 1996

[12] [12]

B. A. Foreman. Connection rules versus differential equations for envelope functions in abrupt heterostructures. Phys. Rev. Lett., 80, 1998

work page 1998

[13] [13]

M. Pistol. Boundary conditions in the effective-mass approximation with a position-dependent mass.Phys. Rev. B, 60, 1999

work page 1999

[14] [14]

Lew Yan Voon

L.C. Lew Yan Voon. Discontinuities in the effective-mass equation.Superlattices and Microstructures, 31(6), 2002

work page 2002

[15] [15]

I. V. Tokatly, A. G. Tsibizov, and A. A. Gorbatsevich. Interface electronic states and boundary conditions for envelope functions.Phys. Rev. B, 65, 2002

work page 2002

[16] [16]

Harrison

W. Harrison. Effects of matching conditions in effective-mass theory: Quantum wells, transmission, and metal- induced gap states.Journal of Applied Physics, 110, 2011

work page 2011

[17] [17]

Gadella, S ¸

M. Gadella, S ¸. Kuru, and J. Negro. Self-adjoint hamiltonians with a mass jump: General matching conditions. Physics Letters A, 362(4), 2007

work page 2007

[18] [18]

Gadella, J

M. Gadella, J. Negro, and L.M. Nieto. Bound states and scattering coefficients of the−aδ(x)+bδ ′(x) potential. Physics Letters A, 373(15), 2009

work page 2009

[19] [19]

P. Kurasov. Distribution theory for discontinuous test functions and differential operators with generalized coefficients.Journal of Mathematical Analysis and Applications, 201(1), 1996

work page 1996

[20] [20]

F. S. A. Cavalcante, R. N. Costa Filho, J. Ribeiro Filho, C. A. S. de Almeida, and V. N. Freire. Form of the quantum kinetic-energy operator with spatially varying effective mass.Phys. Rev. B, 55, 1997

work page 1997

[21] [21]

Mazharimousavi

Omar Mustafa and S. Mazharimousavi. Ordering ambiguity revisited via position dependent mass pseudo- momentum operators.International Journal of Theoretical Physics, 46, 07 2006

work page 2006

[22] [22]

D. J. BenDaniel and C. B. Duke. Space-charge effects on electron tunneling.Phys. Rev., 152, 1966. 15

work page 1966

[23] [23]

Qi-Gao Zhu and H. Kroemer. Interface connection rules for effective-mass wave functions at an abrupt het- erojunction between two different semiconductors.Phys. Rev. B, 27, 1983

work page 1983

[24] [24]

Li and K

Tsung L. Li and K. J. Kuhn. Band-offset ratio dependence on the effective-mass hamiltonian based on a modified profile of the gaas-al xga1−xas quantum well.Phys. Rev. B, 47, 1993

work page 1993

[25] [25]

Gora and F

T. Gora and F. Williams. Theory of electronic states and transport in graded mixed semiconductors.Phys. Rev., 177, 1969

work page 1969

[26] [26]

J. R. F. Lima, M. Vieira, C. Furtado, F. Moraes, and C. Filgueiras. Yet another position-dependent mass quantum model.Journal of Mathematical Physics, 53(7), 2012

work page 2012

[27] [27]

Calogero

F. Calogero. Solution of a three-body problem in one dimension.Journal of Mathematical Physics, 10(12), 1969

work page 1969

[28] [28]

D. Zhu. A new potential with the spectrum of an isotonic oscillator.Journal of Physics A: Mathematical and General, 20(13), 1987

work page 1987

[29] [29]

J. F. Cari˜ nena, A. M. Perelomov, M. F. Ra˜ nada, and M. Santander. A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator.Journal of Physics A: Mathematical and Theoretical, 41(8), 2008

work page 2008

[30] [30]

S. M. Ikhdair and R. Sever. Relativistic and nonrelativistic bound states of the isotonic oscillator by nikiforov- uvarov method.Journal of Mathematical Physics, 52(12), 2011

work page 2011

[31] [31]

J. Sesma. The generalized quantum isotonic oscillator.Journal of Physics A: Mathematical and Theoretical, 43(18), 2010

work page 2010

[32] [32]

V. V. Dodonov, V. I. Man’ko, and L. Rosa. Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential.Phys. Rev. A, 57:2851–2858, Apr 1998

work page 1998

[33] [33]

Dodonov, I.A

V.V. Dodonov, I.A. Malkin, and V.I. Man’ko. Even and odd coherent states and excitations of a singular oscillator.Physica, 72(3):597–615, 1974

work page 1974

[34] [34]

S. M. Nagiyev, E. I. Jafarov, and R. M. Imanov. On a dynamical symmetry group of the relativistic linear singular oscillator.Europhysics Letters, 76(2):175, sep 2006

work page 2006

[35] [35]

Fern´ andez

Francisco M. Fern´ andez. On the quantum-mechanical singular harmonic oscillator. https://arxiv.org/abs/2112.03693, 2023

work page arXiv 2023

[36] [36]

Self-adjoint extension procedure for a singular oscillator

Anzor Khelashvili and Teimuraz Nadareishvili. Self-adjoint extension procedure for a singular oscillator. https://arxiv.org/abs/2406.12927, 2024

work page arXiv 2024

[37] [37]

Fl¨ ugge.Practical Quantum Mechanics

S. Fl¨ ugge.Practical Quantum Mechanics. 0072-7830. Springer, Berlin, Heidelberg, 1999

work page 1999

[38] [38]

Abramowitz and I.A

M. Abramowitz and I.A. Stegun.Handbook of Mathematical Functions: With Formulas, Graphs, and Mathe- matical Tables. Applied mathematics series. Dover Publications, 1965. 16 6 Tables Symmetric Double Heterostructure (2) V(z) m(z) -4 -2 0 2 4 z -5 0 5 10 15 Vin(z) =−µ 2/(1 +z 2), m in(z) =σ 2/(1 +z 2) V0 =V 2 =V in(z0),m 0 =m 2 =m in(z0). σ= 4,µ= 3,z 0 =−z ...

work page 1965