Quantum Systems with jump-discontinuous mass. I

TL;DR

This paper studies a quantum particle with a discontinuous mass profile, revealing complex spectral behavior and multiple semiclassical limits influenced by boundary conditions and mass jumps.

Contribution

It introduces a detailed spectral analysis of quantum systems with jump discontinuities in mass, highlighting the rich interplay between boundary conditions and spectral properties.

Findings

Eigenfunctions show sensitive dependence on energy.

System admits infinitely many semiclassical limits.

Spectral curves are embedded in a two-torus.

Abstract

We consider a free quantum particle in one dimension whose mass profile exhibits jump discontinuities. The corresponding Hamiltonian is a self-adjoint realisation of the kinetic-energy operator, with the specific realisation determined by the boundary conditions at the points of mass discontinuity. For a family of scale-free boundary conditions, we analyse the associated spectral problem. We find that the eigenfunctions exhibit a highly sensitive and erratic dependence on the energy. Notably, the system supports infinitely many distinct semiclassical limits, each labeled by a point on a spectral curve embedded in the two-torus. These results demonstrate a rich interplay between discontinuous coefficients, boundary data, and spectral asymptotics.