Propagators of singular anharmonic oscillators with quasi-equidistant spectra

TL;DR

The paper tackles the challenge of obtaining exact time-dependent propagators for one-dimensional singular and rationally extended oscillators. It employs Nth-order Darboux transformations of the harmonic oscillator and an image-method regularization to handle 1/x^2 type singularities, yielding closed-form propagators. It identifies two- and three-well rational extensions, analyzes their quasi-equidistant spectra, and provides a physical embedding in three-dimensional problems with axially symmetric magnetic fields and Aharonov-Bohm flux, linking radial dynamics to these solvable models. The results offer analytical tools for studying dynamics and vortex-like states in singular quantum systems, with supplementary materials for reproducibility.

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.

Figures (5)

• Figure 1: Level diagram of Darboux transformations from the Harmonic oscillator (HO) to the sigular Harmonic oscillator (SHO) and its rational quasi-isospectral deformations. Dotted lines show HO eigenvalues, thin black lines show eigenvalues of $H^{\{1\}}$. A possible gap sequence of an arbitrary Darboux transformation are shown by thick black segments.
• Figure 2: Examples of potentials
• Figure 3: Transition probabilities $|K^{\{1,6,7\}}(x,y,t)|^2$ for the potential $V^{\{1,6,7\}}$
• Figure 4: Transition probabilities $|K^{\{1,8,9\}}(x,y,t)|^2$ for the potential $V^{\{1,8,9\}}$
• Figure 5: Radial potential $V^{1,6,7}(\rho)$ and corresponding magnetic fields $B_z(\rho,E_{reg})$ which produce isopectral $l=1$ radial states.