Quantum solvability of a nonlinear $\delta$-type mass profile system: Coupling constant quantization

TL;DR

This paper investigates the quantum behavior of a nonlinear, position-dependent mass system with a singular mass profile, revealing that the coupling constant becomes quantized, confirmed through both quantum and semiclassical analyses.

Contribution

It introduces a general ordered Hamiltonian with a singular mass profile and demonstrates the quantization of the coupling constant in this nonlinear quantum system.

Findings

Quantum solutions are bounded despite the singular mass profile.

The coupling constant is quantized in the system.

Semiclassical analysis confirms the quantum results.

Abstract

In this paper, we discuss the quantum dynamics of a nonlinear system that admits temporally localized solutions at the classical level. We consider a general ordered position-dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. The mass function here is singular at the origin. We observe that the quantum system admits bounded solutions but importantly the coupling parameter of the system gets quantized which has also been confirmed by the semiclassical study as well.