Geometric Phase in Quantum Billiards with a Pointlike Scatterer

TL;DR

This paper investigates how a pointlike scatterer affects quantum energy levels and Berry phases in rectangular billiards, revealing degeneracies, localized states, and sign reversals in one dimension.

Contribution

It introduces a systematic analysis of degeneracies and Berry phases caused by a pointlike scatterer in quantum billiards, including novel findings on localized states and ground state sign reversal.

Findings

Identification of diabolical degeneracies among energy levels.

Illustration of Berry phase associated with degeneracies.

Discovery of localized wave functions and sign reversal in the ground state.

Abstract

We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.