Isochronous oscillator with a singular position-dependent mass and its quantization

TL;DR

This paper analyzes a singular position-dependent-mass oscillator, solves its classical dynamics, and performs a canonical quantization showing its spectrum matches that of the isotonic oscillator with equally spaced energy levels.

Contribution

It introduces a solvable isochronous oscillator with a singular mass profile and demonstrates its quantum equivalence to the isotonic oscillator, including spectral properties.

Findings

Classical dynamics are exactly solvable for both branches of the singular potential.

Quantum spectrum is infinite and equally spaced, similar to the isotonic oscillator.

Energy level spacing is unaffected by ordering ambiguities in the quantization process.

Abstract

In this paper, we present an analysis of the equation $\ddot{x} - (1/2x) \dot{x}^2 + 2 \omega^2 x - 1/8x = 0$, where $\omega > 0$ and $x = x(t)$ is a real-valued variable. We first discuss the appearance of this equation from a position-dependent-mass scenario in which the mass profile goes inversely with $x$, admitting a singularity at $x = 0$. The associated potential is also singular at $x = 0$, splitting the real axis into two halves, i.e., $x > 0$ and $x < 0$. The dynamics is exactly solvable for both the branches and so for definiteness, we stick to the $x > 0$ branch. Performing a canonical quantization in the position representation and upon employing the ordering strategy of the kinetic-energy operator due to von Roos, we show that the problem is isospectral to the isotonic oscillator. Thus, the quantum spectrum consists of an infinite number of equispaced levels. The spacing between the energy levels is found to be insensitive to the specific choices of the ambiguity parameters that are employed for ordering the kinetic-energy operator \`a la von Roos.