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arxiv: 2606.22107 · v2 · pith:ZHCYVPS5new · submitted 2026-06-20 · 🪐 quant-ph

Spectral and thermodynamic properties of supersymmetric quantum systems with self-adjoint deformed momentum

J. A. Oke , F. A. Dossa This is my paper

Pith reviewed 2026-07-01 06:51 UTC · model grok-4.3

classification 🪐 quant-ph
keywords supersymmetric quantum mechanicsdeformed momentum operatorself-adjoint operatorsdensity of statesthermodynamic propertiesheat capacitygeometric deformationpartition function
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The pith

Geometric deformations of momentum operators produce distinct heat capacity signatures in supersymmetric quantum systems.

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

The paper builds a framework that makes deformed momentum operators strictly self-adjoint using the generalized extended momentum operator formalism inside a non-Hermitian supersymmetric factorization scheme. This construction works uniformly for linear and quadratic position-dependent deformations without separate boundary-condition adjustments and delivers exact analytical energy spectra together with hidden su(1,1) symmetry. Thermodynamic quantities are then obtained from the partition function via the Euler-Maclaurin formula, showing that the altered density of states produces qualitatively different heat-capacity curves: a divergent peak for the linear case and saturation below the classical limit for the quadratic case.

Core claim

The GEMO formalism guarantees intrinsic self-adjointness for both linear (μ(x)=αx) and quadratic (μ(x)=αx²) deformations inside a unified supersymmetric scheme, yielding exact spectra and partition functions whose associated density of states ρ(E) produces a divergent heat-capacity peak for linear deformation and saturation C/k_B → 0.6 for quadratic deformation.

What carries the argument

The generalized extended momentum operator (GEMO) formalism, which enforces intrinsic self-adjointness of the deformed momentum inside a non-Hermitian supersymmetric factorization scheme.

If this is right

  • Exact analytical spectra become available for supersymmetric systems with these deformations.
  • The linear deformation produces a divergent heat-capacity peak caused by accumulation of states at a finite maximum energy.
  • The quadratic deformation produces a heat capacity that saturates at C/k_B ≈ 0.6, below the Dulong-Petit value.
  • Geometric deformation functions as a tunable control parameter for the thermodynamic response.

Where Pith is reading between the lines

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

  • The same construction may supply solvable models for effective-mass or curved-space nanostructures.
  • Comparison of the two deformation classes could guide selection of position-dependent mass profiles in device design.
  • The hidden su(1,1) symmetry might allow algebraic computation of higher thermodynamic moments without explicit summation.

Load-bearing premise

The GEMO formalism automatically supplies intrinsic self-adjointness for both chosen deformations without any additional boundary-condition requirements.

What would settle it

Measurement of heat capacity versus temperature in a physical realization of a linear-deformation system that shows a divergent peak near the maximal energy, or in a quadratic-deformation system that saturates near 0.6 k_B, would confirm or refute the thermodynamic predictions.

Figures

Figures reproduced from arXiv: 2606.22107 by F. A. Dossa, J. A. Oke.

Figure 1
Figure 1. Figure 1: illustrates the evolution of energy levels as a function of the deformation parameter 𝛼. In the linear case figure (a), we observe that for low values of |𝛼|, all excited energy levels remain positive, while the fundamental level is strictly positive regardless of the value of 𝛼. This suggests a stability of the spectrum for small deformations. On the other hand, in the quadratic case figure (b), all energ… view at source ↗
Figure 2
Figure 2. Figure 2: (Colour online) Thermodynamic properties as a function of temperature for different values of 𝛼 (case of linear deformation). For the quadratic deformation 𝜇(𝑥) = 𝛼𝑥2 , the coefficient 𝐴2 = 2ℏ 2𝛼/𝑚 > 0 (𝛼 > 0) induces a spectrum that grows quadratically, 𝐸𝑛 ∝ 𝑛 2 , for large 𝑛, which modifies the density of states to 𝜌𝛼(𝐸) ∝ 1/ √ 𝐸. This decay with energy limits the number of thermally accessible states at… view at source ↗
Figure 3
Figure 3. Figure 3: (Colour online) Thermodynamic properties as a function of temperature for different values of 𝛼 (case of quadratic deformation). Finally, the entropy (𝑆) continues to increase with temperature, but its growth is significantly slowed in the presence of a quadratic deformation. This attenuation reflects a more gradual access to disordered states, consistent with a narrower energy spectrum for 𝛼 > 0. In concl… view at source ↗
read the original abstract

We establish a rigorous framework for quantum systems with geometric deformations by constructing a strictly self-adjoint deformed momentum operator through the generalized extended momentum operator (GEMO) formalism. Unlike previous approaches relying on boundary-condition hermiticity, our method ensures intrinsic self-adjointness for both linear ($\mu(x)=\alpha x$) and quadratic ($\mu(x)=\alpha x^{2}$) deformations within a unified non-Hermitian supersymmetric factorization scheme. This yields exact analytical spectra while revealing hidden $\mathfrak{su}(1,1)$ symmetry structures. Crucially, we provide the first complete thermodynamic characterization of such systems by analytically evaluating the partition function via the Euler--Maclaurin approximation. Geometric deformation fundamentally reshapes the density of states $\rho(E)$, producing distinct thermal signatures: a divergent heat capacity peak for linear deformation due to state accumulation near a maximal energy, and a saturation $C/k_{\mathrm{B}}\to 0.6$ (below the Dulong--Petit limit) for quadratic deformation. These results establish geometric deformation as a tunable parameter for engineering quantum thermodynamic responses in curved nanostructures.

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

3 major / 2 minor

Summary. The paper introduces the generalized extended momentum operator (GEMO) formalism to construct strictly self-adjoint deformed momentum operators for linear (μ(x)=αx) and quadratic (μ(x)=αx²) geometric deformations within a unified non-Hermitian supersymmetric factorization scheme. This is claimed to yield exact analytical spectra exhibiting hidden su(1,1) symmetry, without relying on boundary-condition adjustments. The partition function is then evaluated analytically via the Euler-Maclaurin approximation to obtain the density of states ρ(E) and thermodynamic quantities, revealing a divergent heat-capacity peak for linear deformation (due to state accumulation near a maximal energy) and saturation of C/k_B to 0.6 (below the Dulong-Petit value) for quadratic deformation.

Significance. If the self-adjointness construction and exact spectra are rigorously established, the work would provide a tunable geometric parameter for engineering distinct thermodynamic responses in quantum systems, with potential relevance to curved nanostructures. The unified treatment of two deformation types and the identification of su(1,1) symmetry are positive features. However, the central claims rest on unverified aspects of the GEMO construction and the approximation step, limiting the current impact.

major comments (3)
  1. [GEMO formalism] The GEMO formalism section: the claim of intrinsic self-adjointness for both deformations via non-Hermitian SUSY factorization without boundary adjustments is load-bearing for the exact real spectra and subsequent thermodynamics, yet no domain specification, deficiency-index calculation, or explicit verification that the operator is essentially self-adjoint on the chosen dense domain is provided; this directly undermines the asserted exact analytical spectra and the derived divergent/saturation behaviors of C.
  2. [Thermodynamic characterization] The Euler-Maclaurin step for the partition function: no error estimates or convergence analysis for the approximation are supplied, which is required to support the quantitative claims such as C/k_B → 0.6 for quadratic deformation and the divergent peak for linear deformation.
  3. [Spectra and symmetry] Abstract and § on spectra: the assertion of 'exact analytical spectra' for both deformations is not accompanied by the explicit eigenvalue expressions or the demonstration that the non-Hermitian factorization closes the operator without implicit boundary conditions, making the thermal signatures (state accumulation near maximal energy, saturation below Dulong-Petit) rest on an unproven foundation.
minor comments (2)
  1. Notation for the deformation parameter α and the function μ(x) should be introduced with a clear table or equation early in the manuscript for consistency across sections.
  2. The abstract states 'the first complete thermodynamic characterization'; this should be qualified with references to prior partial studies on deformed systems if they exist.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment point by point below, indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [GEMO formalism] The GEMO formalism section: the claim of intrinsic self-adjointness for both deformations via non-Hermitian SUSY factorization without boundary adjustments is load-bearing for the exact real spectra and subsequent thermodynamics, yet no domain specification, deficiency-index calculation, or explicit verification that the operator is essentially self-adjoint on the chosen dense domain is provided; this directly undermines the asserted exact analytical spectra and the derived divergent/saturation behaviors of C.

    Authors: We agree that an explicit deficiency-index calculation is not included in the current manuscript and would strengthen the self-adjointness claim. The GEMO construction ensures symmetry on the dense domain of compactly supported smooth functions, with the non-Hermitian SUSY factorization guaranteeing real spectra through the partner Hamiltonian. In the revised version we will add a dedicated subsection specifying the domain and proving essential self-adjointness (deficiency indices (0,0)) via the bounded deformation-induced properties. revision: yes

  2. Referee: [Thermodynamic characterization] The Euler-Maclaurin step for the partition function: no error estimates or convergence analysis for the approximation are supplied, which is required to support the quantitative claims such as C/k_B → 0.6 for quadratic deformation and the divergent peak for linear deformation.

    Authors: We acknowledge that explicit error estimates and convergence analysis for the Euler-Maclaurin approximation are absent. The approximation is used to evaluate the partition function from the density of states, and the leading-order terms capture the reported qualitative behaviors. We will incorporate remainder bounds and convergence analysis in the revision, leveraging the specific forms of ρ(E) for each deformation to support the quantitative thermodynamic claims. revision: yes

  3. Referee: [Spectra and symmetry] Abstract and § on spectra: the assertion of 'exact analytical spectra' for both deformations is not accompanied by the explicit eigenvalue expressions or the demonstration that the non-Hermitian factorization closes the operator without implicit boundary conditions, making the thermal signatures (state accumulation near maximal energy, saturation below Dulong-Petit) rest on an unproven foundation.

    Authors: The manuscript derives the explicit eigenvalue expressions in the spectra section from the SUSY partner equations and identifies the su(1,1) generators. The factorization closes without boundary conditions because the superpotential satisfies the Riccati equation exactly, producing square-integrable solutions. We will revise by adding the explicit expressions to the abstract and expanding the closure argument with a dedicated paragraph to make these steps fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: GEMO construction and Euler-Maclaurin thermodynamics are independent of the reported spectra and heat-capacity signatures.

full rationale

The provided abstract and reader summary contain no equations, fitted parameters, or self-citations that reduce the claimed spectra, density of states, or thermodynamic quantities (divergent C peak, C/k_B → 0.6) to tautological redefinitions of the deformation parameter α or the GEMO formalism itself. The partition function is evaluated via the standard Euler-Maclaurin approximation applied to analytically obtained spectra; no step is shown to be a statistical fit renamed as a prediction, nor does any uniqueness theorem or ansatz reduce to prior self-authored work by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claims rest on the unproven assertion that GEMO produces intrinsic self-adjointness for the chosen deformations and on the applicability of the Euler-Maclaurin formula to the resulting discrete spectrum without stated error bounds.

free parameters (1)
  • deformation strength α
    The functions μ(x)=αx and μ(x)=αx² introduce α as a free scale that controls the strength of the geometric deformation and therefore enters all spectra and thermodynamic quantities.
axioms (2)
  • domain assumption The generalized extended momentum operator (GEMO) formalism guarantees intrinsic self-adjointness for linear and quadratic deformations inside a non-Hermitian supersymmetric factorization scheme.
    Invoked to replace boundary-condition hermiticity and to obtain exact spectra.
  • domain assumption The Euler-Maclaurin formula yields an accurate analytic approximation to the partition function for the deformed spectra.
    Used to obtain closed-form thermodynamic quantities.
invented entities (1)
  • GEMO formalism no independent evidence
    purpose: To construct a strictly self-adjoint deformed momentum operator
    New formalism introduced to achieve intrinsic self-adjointness without boundary conditions.

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