Quantum pseudo-integrable Hamiltonian impact systems

TL;DR

This paper introduces a quantization approach for a pseudointegrable Hamiltonian impact system, analyzing energy levels, wavefunctions, and their statistical properties, revealing persistent wavefunction concentration unlike typical equidistribution.

Contribution

It presents the first quantization of a pseudointegrable impact system, including EBK conditions, Weyl's law verification, and analysis of wavefunction behavior at high energies.

Findings

Energy level statistics resemble pseudointegrable billiards

Wavefunction concentration persists at high energies

No complete equidistribution of wavefunctions in configuration space

Abstract

Quantization of a toy model of a pseudointegrable Hamiltonian impact system is introduced, including EBK quantization conditions, a verification of Weyl's law, the study of their wavefunctions and a study of their energy levels properties. It is demonstrated that the energy levels statistics are similar to those of pseudointegrable billiards. Yet, here, the density of wavefunctions which concentrate on projections of classical level sets to the configuration space does not disappear at large energies, suggesting that there is no equidistribution in the configuration space in the large energy limit; this is shown analytically for some limit symmetric cases and is demonstrated numerically for some nonsymmetric cases.