Isospectrality and non-locality of generalized Dirac combs

TL;DR

This paper explores a generalized Dirac comb model with a four-parameter family of point interactions, revealing complex spectral relationships and classifying isospectral Hamiltonians, including their symmetries and spectral uniqueness.

Contribution

It introduces a comprehensive analysis of the spectral properties of generalized point interactions, extending Dirac comb models and classifying isospectral relations.

Findings

Identified a rich structure of isospectrality among generalized point interactions.

Classified which Hamiltonians are spectrally unique and which are related by symmetries.

Mapped the parameter dependence of spectral properties in the model.

Abstract

We consider a generalization of Dirac's comb model, describing a non-relativistic particle moving in a periodic array of generalized point interactions. The latter represent the most general point interactions rendering the kinetic-energy operator self-adjoint, and form a four-parameters family that includes the $\delta$-potential and the $\delta'$-potential as particular cases. We study the parameter dependence of the spectral properties of this system, finding a rich isospectrality structure. We systematically classify a large class of isospectral relations, determining which Hamiltonians are spectrally unique, and which are instead related by a unitary or anti-unitary transformation.