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arXiv:2512.16695

Propagators of singular anharmonic oscillators with quasi-equidistant spectra

Andrey M. Pupasov-Maksimov, Marcelo Silva Oliveira

quant-phmath-phSubmitted Dec 18, 2025• Announced Dec 19, 2025
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Abstract

Darboux transformations of the singular harmonic oscillator are considered. Analytical expressions for the propagators are obtained, using the image method applied to formal singular propagators. Two-well and three-well families of potentials and the corresponding propagators are presented. Axially symmetric magnetic field configurations corresponding to these potentials have been identified.

Comments
20 pages, 5 figures

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Darboux transformations of the singular harmonic oscillator are considered. Analytical expressions for the propagators are obtained, using the image method applied to formal singular propagators. Two-well and three-well families of potentials and the corresponding propagators are presented. Axially symmetric magnetic field configurations corresponding to these potentials have been identified.
Two kinds of transformation for the time-dependent Schrödinger equation, i.e. the differential and integral transformations, are introduced. If one considers only stationary solutions of this equation, both transformations reduce to the well known Darboux transformation for the stationary Schrödinger equation. When applied to non-stationary solutions, they give different results. Both transformations are invertible in appropriate spaces. With the help of these transformations alternative systems of coherent states to those in the literature are obtained for isospectral Hamiltonians with equidistant spectra. These transformations are also applied to the construction of coherent states for Hamiltonians whose spectrum consists of an equidistant part and one separately disposed level with an energy gap equal to the k skipped levels.
In this paper we solve exactly the problem of the spectrum and Feynman propagator of a charged particle submitted to both an anharmonic oscillator in the plane and a constant and homogeneous magnetic field of arbitrary strength aligned with the perpendicular direction to the plane. As we shall see in the beginning of the letter, the Hamiltonian, being a quadratic form, is easily diagonalizable and the Classical Action can be used to construct the exact Feynman Propagator using the Stationary Phase Approximation. The result is useful for the treatment of quasi two dimensional samples in the field of magnetic effects in nano-structures and quantum optics. The presented solution, after minor extensions, can also be used for motion in three dimensions, and in fact it has been used for years in such cases. Also it can be used as a good exercise of a Feynman Path Integral that can be calculated easily with just the help of the Classical Action.
Quantum mechanical anharmonic oscillators and Hamiltonians for particles in external magnetic fields are related to representations of nilpotent groups. Using this connection the eigenfunctions of the quartic anharmonic oscillator with potential Vα(x)=(α+(x2/2))2 can be used to determine the eigenfunctions of a charged particle in a nonconstant magnetic field, of the form Bz=β2+β3x. The quartic anharmonic oscillator eigenvalues for low-lying states are obtained numerically and a function which interpolates between α≪0 (a double harmonic oscillator) and α≫0 (a harmonic oscillator) is shown to give a good fit to the numerical data. Approximate expressions for the quartic anharmonic oscillator eigenfunctions are then used to get the eigenfunctions for the magnetic field Hamiltonian.