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arxiv: 2605.01443 · v1 · submitted 2026-05-02 · 🧮 math-ph · math.MP· quant-ph

A position dependent mass Hamiltonian and abstract ladder operators

Fabio Bagarello , Emanuele Balistreri , Antonino Faddetta This is my paper

Pith reviewed 2026-05-08 19:25 UTC · model grok-4.3

classification 🧮 math-ph math.MPquant-ph
keywords position-dependent massabstract ladder operatorspseudo-bosonic operatorsbi-coherent statesnon-self-adjoint Hamiltonianfactorizable operatorsquantum mechanics
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The pith

Abstract ladder operators find eigenvalues and eigenvectors for a position-dependent mass Hamiltonian that is factorizable, even if not self-adjoint, via pseudo-bosonic operators and attached bi-coherent states.

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

The paper applies the method of abstract ladder operators to a one-dimensional Hamiltonian for a particle whose mass varies with position. It restricts attention to factorizable cases and drops the usual self-adjointness requirement. Under these conditions the ladder operators generate pseudo-bosonic operators whose action yields the energy eigenvalues and eigenvectors. The same operators allow the explicit construction of bi-coherent states. A reader would care because the setting covers physically relevant models in which mass is not constant and the operator may fail to be Hermitian.

Core claim

We consider the Hamiltonian H of a particle in one dimension with a position dependent mass for which we apply the recent strategy of the so-called abstract ladder operators, in the attempt to find its eigenvalues and eigenvectors. We don't assume that H is self-adjoint, while we focus on the case of a factorizable operator. We show then that pseudo-bosonic operators play a relevant role in this analysis, and we construct bi-coherent states attached to these operators.

What carries the argument

Abstract ladder operators defined on a factorizable position-dependent-mass Hamiltonian, which produce pseudo-bosonic operators and permit construction of bi-coherent states.

If this is right

  • The eigenvalues of such Hamiltonians can be found algebraically without solving a differential equation.
  • The corresponding eigenvectors are obtained by applying the ladder operators to a reference state.
  • Bi-coherent states exist and are explicitly constructible for the pseudo-bosonic operators that arise.
  • The procedure remains valid for non-self-adjoint operators that are still factorizable.

Where Pith is reading between the lines

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

  • The same ladder-operator technique may apply to other non-Hermitian Hamiltonians that arise in effective models with variable coefficients.
  • Bi-coherent states could be used to study time evolution or resolution of the identity in position-dependent-mass systems.
  • The construction suggests a route for defining coherent-state bases in any quantum system whose Hamiltonian is factorizable but not necessarily Hermitian.

Load-bearing premise

The Hamiltonian must admit a factorization that lets abstract ladder operators be defined and applied to extract the spectrum.

What would settle it

A concrete factorizable position-dependent-mass Hamiltonian whose eigenvalues or eigenvectors cannot be recovered by repeated action of the abstract ladder operators.

Figures

Figures reproduced from arXiv: 2605.01443 by Antonino Faddetta, Emanuele Balistreri, Fabio Bagarello.

Figure 1
Figure 1. Figure 1: |ψ(z; x)| 2 (in orange) and |α(z, λ)φ(z1; x)| 2 (in blue) for m(x) = m0e −x 2 . On the top we fix x = 1 and λ = −2 and we plot the functions for zr, zi , and γ = 1 (top left) and γ = 1+2i (top right). Bottom: we plot the functions for x, zi and zR = 3, λR = −2, γ = 1 with λ = −2 (left) and λ = −2 + 2i (right). and it follows from (C.5) Nφ,0 = Nψ,0 =  ℏ r π −λ e − γ 2ℏ 2 λ − 1 2 . In this case the square-… view at source ↗
Figure 2
Figure 2. Figure 2: depicts the same difference already observed in view at source ↗
Figure 3
Figure 3. Figure 3: shows again the same differences beween self-adjoint and non self-adjoint cases for bi-coherent states view at source ↗
read the original abstract

We consider the Hamiltonian $H$ of a particle in one dimension with a position dependent mass for which we apply the recent strategy of the so-called {\em abstract ladder operators}, in the attempt to find its eigenvalues and eigenvectors. We don't assume that $H$ is self-adjoint, while we focus on the case of a factorizable operator. We show then that pseudo-bosonic operators play a relevant role in this analysis, and we construct bi-coherent states attached to these operators. Explicit examples are discussed.

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

0 major / 3 minor

Summary. The manuscript considers a one-dimensional Hamiltonian with position-dependent mass that is factorizable (but not necessarily self-adjoint). It applies the abstract ladder operator technique to determine eigenvalues and eigenvectors, establishes the relevance of pseudo-bosonic operators in this setting, constructs associated bi-coherent states, and illustrates the results with explicit examples.

Significance. If the constructions are correct, the work extends abstract ladder and pseudo-boson methods to a physically relevant class of non-Hermitian position-dependent-mass operators. The explicit examples provide concrete verification and could facilitate applications in variable-mass quantum models. The approach is grounded in the factorizability assumption rather than ad-hoc fitting.

minor comments (3)
  1. [Abstract] The abstract states that explicit examples are discussed, but the introduction or §2 could briefly name the specific forms of the mass function m(x) and potential used in those examples to orient the reader.
  2. [§3] Notation for the abstract ladder operators A and A† (and their pseudo-bosonic counterparts) should be introduced with a short comparison table or explicit commutation relations to avoid confusion with standard bosonic operators.
  3. [§4] In the construction of bi-coherent states, the overlap or resolution-of-identity property is asserted; a one-line reference to the relevant prior result (or a short derivation) would strengthen readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for recommending minor revision. The referee's description accurately reflects the scope and contributions of the work on abstract ladder operators for a factorizable position-dependent mass Hamiltonian, including the role of pseudo-bosonic operators and bi-coherent states.

Circularity Check

0 steps flagged

Minor self-citation of abstract ladder/pseudo-boson framework; central application to factorizable PDM Hamiltonian remains independent

full rationale

The paper applies the abstract ladder operator method to a position-dependent-mass Hamiltonian that is factorizable (but not necessarily self-adjoint), derives the relevance of pseudo-bosonic operators, and constructs attached bi-coherent states, with explicit examples provided. These techniques originate in prior literature that overlaps with the authors, but the load-bearing steps are the explicit factorization assumption on H and the direct construction of eigenvalues, eigenvectors, and states for the new Hamiltonian class. No step reduces by definition to a fitted parameter, self-referential renaming, or unverified self-citation chain; the result is a concrete application rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; no explicit free parameters, axioms, or invented entities are identifiable. The factorizability of the operator and the existence of abstract ladder operators are invoked but not detailed.

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