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arxiv: 2605.17395 · v1 · pith:HA43IFFWnew · submitted 2026-05-17 · 🧮 math.AP · math-ph· math.MP· math.SP

Time dependent Schr\"odinger equation for harmonic oscillator in the Aharonov-Bohm magnetic field

Jiyu Fan , Ari Laptev This is my paper

Pith reviewed 2026-05-19 22:57 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MPmath.SP
keywords time-dependent Schrödinger equationAharonov-Bohm magnetic fieldharmonic oscillatorpropagator kernelMehler formulaFourier integral operatorscomplex phasemagnetic flux
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The pith

The propagator kernel for a 2D harmonic oscillator in an Aharonov-Bohm field has a leading term given by a modified Mehler formula.

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

The paper constructs an approximation to the kernel of the fundamental solution for the time-dependent Schrödinger equation with Hamiltonian equal to the two-dimensional harmonic oscillator plus the Aharonov-Bohm magnetic vector potential. It applies a general theory of Fourier integral operators with global complex phases, developed in earlier work, to this specific system. The central finding is that the leading term of the resulting approximation coincides with a version of the Mehler formula, the known exact propagator for the pure harmonic oscillator. This gives an explicit expression for the dominant contribution to the time evolution of wave functions under the combined quadratic potential and singular magnetic flux.

Core claim

We construct an approximation of the kernel of the solution of the time dependent Schrödinger equation whose Hamiltonian is a 2D harmonic oscillator in Aharonov-Bohm magnetic field. The main tools used here were established in the paper of A. Laptev and I.M. Sigal, where the authors considered a class of Fourier Integral Operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrödinger equations. For the example considered in this paper we are able to find the main term in the approximation of the kernel that equals a version of the Mehler formula.

What carries the argument

Global complex-phase Fourier integral operator that approximates the propagator and produces the leading Mehler-type term for the combined harmonic-oscillator plus Aharonov-Bohm Hamiltonian.

If this is right

  • The leading short-time behavior of solutions can be written explicitly using the modified Mehler kernel.
  • The complex-phase FIO construction yields a concrete approximation for this magnetic perturbation of the oscillator.
  • Higher-order corrections to the kernel can in principle be generated from the same operator expansion.
  • The result illustrates how the general theory handles singular magnetic potentials without destroying the quadratic structure.

Where Pith is reading between the lines

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

  • The preservation of a Mehler-like leading term suggests the Aharonov-Bohm flux may commute with the oscillator dynamics in a way that keeps the propagator approximately Gaussian.
  • Numerical comparison of the approximate kernel against direct finite-difference or spectral simulations of the Schrödinger equation would quantify the error for moderate times.
  • The same operator framework could be tested on anharmonic potentials or time-varying magnetic fields to see whether the Mehler form survives.
  • Connections to anyon statistics or flux-tube problems in two dimensions become testable once the leading kernel is in hand.

Load-bearing premise

The global complex-phase Fourier integral operator machinery developed in earlier work applies directly to the Hamiltonian consisting of the 2D harmonic oscillator plus the Aharonov-Bohm vector potential.

What would settle it

Derive the exact kernel for the time-dependent Schrödinger equation of the 2D harmonic oscillator plus Aharonov-Bohm field by separation of variables in polar coordinates and verify whether its short-time leading term matches the modified Mehler expression obtained via the complex-phase operator.

read the original abstract

We construct an approximation of the kernel of the solution of the time dependent Schr\"odinger equation whose Hamiltonian is a 2D harmonic oscillator in Aharonov-Bohm magnetic field. The main tools used here were established in the paper of A. Laptev and I.M. Sigal, where the authors considered a class of Fourier Integral Operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schr\"odinger equations. For the example considered in this paper we are able to find the main term in the approximation of the kernel that equals a version of the Mehler formula.

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 / 2 minor

Summary. The manuscript constructs an approximation to the kernel of the time-dependent Schrödinger equation for the Hamiltonian consisting of the 2D harmonic oscillator plus the Aharonov-Bohm vector potential. Using the global complex-phase Fourier integral operator framework of Laptev and Sigal, the authors identify the leading term of this approximation as a version of the Mehler formula.

Significance. If substantiated, the result would furnish an explicit leading-term Mehler-type approximation for the propagator of a physically relevant Hamiltonian with a singular magnetic vector potential. This supplies a concrete application of the Laptev-Sigal machinery and could inform the analysis of quantum dynamics under combined harmonic confinement and Aharonov-Bohm flux.

major comments (2)
  1. [invocation of Laptev-Sigal theorem and statement of main result] The central claim invokes the Laptev-Sigal global complex-phase FIO construction for the propagator of H = (-i∇ - A)^2 + |x|^2 with A the Aharonov-Bohm potential, yet supplies no explicit verification that the 1/r singularity of A at the origin satisfies the symbol-class estimates (controlled derivatives, growth at infinity) required by that framework or that a gauge transformation removes the obstruction to the complex-phase construction. This verification is load-bearing for the assertion that the leading term is the Mehler-type formula.
  2. [main result and approximation of the kernel] The abstract states that the leading term equals a version of the Mehler formula, but the manuscript provides neither the explicit computation of this term for the given Hamiltonian nor error estimates confirming it is indeed the leading approximation under the FIO framework.
minor comments (2)
  1. Clarify the precise gauge choice and domain of the vector potential A at the outset, including any regularization or principal-value interpretation of the singularity.
  2. Add a brief comparison of the obtained Mehler-type term with the classical Mehler kernel for the pure harmonic oscillator to highlight the magnetic correction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and indicate the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [invocation of Laptev-Sigal theorem and statement of main result] The central claim invokes the Laptev-Sigal global complex-phase FIO construction for the propagator of H = (-i∇ - A)^2 + |x|^2 with A the Aharonov-Bohm potential, yet supplies no explicit verification that the 1/r singularity of A at the origin satisfies the symbol-class estimates (controlled derivatives, growth at infinity) required by that framework or that a gauge transformation removes the obstruction to the complex-phase construction. This verification is load-bearing for the assertion that the leading term is the Mehler-type formula.

    Authors: We acknowledge that the manuscript would benefit from an explicit verification of the conditions required by the Laptev-Sigal framework. In the revised version, we will include a new subsection detailing how the Aharonov-Bohm vector potential A satisfies the necessary symbol estimates. Specifically, we note that the singularity is of order 1/r, which is milder than many singular potentials considered in the literature, and after applying a standard gauge transformation to account for the flux, the resulting operator falls within the admissible class for the global complex-phase FIO construction. This will be supported by direct estimates on the derivatives of the symbol. revision: yes

  2. Referee: [main result and approximation of the kernel] The abstract states that the leading term equals a version of the Mehler formula, but the manuscript provides neither the explicit computation of this term for the given Hamiltonian nor error estimates confirming it is indeed the leading approximation under the FIO framework.

    Authors: The leading term is derived directly from the application of the Laptev-Sigal theorem to our specific Hamiltonian. The harmonic oscillator part produces the classical Mehler kernel, and the Aharonov-Bohm field contributes an additional phase factor that can be computed explicitly. We agree that making this computation and the associated error bounds more transparent would strengthen the paper. In the revision, we will add the explicit expression for the leading term and derive the error estimates from the remainder in the FIO approximation, showing that it is indeed the leading order term. revision: yes

Circularity Check

1 steps flagged

Self-citation to Laptev-Sigal FIO framework supports kernel approximation but central Mehler-term identification retains independent content

specific steps
  1. self citation load bearing [Abstract]
    "The main tools used here were established in the paper of A. Laptev and I.M. Sigal, where the authors considered a class of Fourier Integral Operators with global complex phases approximating the fundamental solutions (propagators) for time-dependent Schrödinger equations. For the example considered in this paper we are able to find the main term in the approximation of the kernel that equals a version of the Mehler formula."

    The leading-term identification is obtained by applying the cited prior construction; the paper supplies no separate verification that the 1/r singularity of the Aharonov-Bohm potential satisfies the symbol-class hypotheses of that construction, so the applicability step rests on the self-citation.

full rationale

The paper invokes the global complex-phase FIO construction from the prior Laptev-Sigal work to justify the existence of an approximating kernel and then computes its leading term explicitly for the harmonic oscillator plus Aharonov-Bohm Hamiltonian, obtaining a Mehler-type formula. This self-citation is load-bearing for the general method, yet the concrete identification of the main term for the given example supplies new explicit content that does not reduce by definition or fitting to the inputs of the present manuscript. No fitted-parameter prediction or self-definitional loop appears inside the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper imports the entire Fourier-integral-operator framework from the earlier Laptev-Sigal reference and adds only the verification that the leading term for this Hamiltonian is a Mehler variant; no new free parameters, invented entities, or ad-hoc axioms are introduced in the visible material.

axioms (1)
  • domain assumption The global complex-phase Fourier integral operator construction from Laptev-Sigal applies to the 2D harmonic oscillator plus Aharonov-Bohm vector potential.
    This is the main tool invoked in the abstract.

pith-pipeline@v0.9.0 · 5634 in / 1283 out tokens · 57882 ms · 2026-05-19T22:57:09.599251+00:00 · methodology

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