Hearing the shape of a quantum boundary condition

TL;DR

This paper investigates the spectral properties of a quantum particle on a ring with a junction, classifying all self-adjoint boundary conditions and identifying isospectral Hamiltonian families based on symmetry considerations.

Contribution

It provides a complete classification of self-adjoint boundary conditions for the quantum ring with a junction and characterizes isospectral Hamiltonians using spectral functions and symmetry analysis.

Findings

Classified all self-adjoint realizations of the Hamiltonian.

Identified two classes of isospectral Hamiltonians distinguished by parity symmetry.

Connected spectral properties to boundary conditions and symmetry actions.

Abstract

We study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion.