Duality of bounded and scattering wave systems with local symmetries

TL;DR

This paper explores the duality between bounded and scattering wave systems with local symmetries, revealing how eigenenergy extrema relate to perfect transmission resonances and local symmetry eigenstates.

Contribution

It introduces a novel connection between spectral properties of bounded systems and scattering resonances in systems with local symmetries, supported by numerical examples.

Findings

Eigenenergy extrema correspond to perfect transmission resonances.

Wavefunctions become eigenstates of local symmetry transforms.

Duality links scattering resonances with bounded system eigenstates.

Abstract

We investigate the spectral properties of a class of hard-wall bounded systems, described by potentials exhibiting domain-wise different local symmetries. Tuning the distance of the domains with locally symmetric potential from the hard wall boundaries leads to extrema of the eigenenergies. The underlying wavefunction becomes then an eigenstate of the local symmetry transform in each of the domains of local symmetry. These extrema accumulate towards eigenenergies which do not depend on the position of the potentials inside the walls. They correspond to perfect transmission resonances of the associated scattering setup, obtained by removing the hard walls. We argue that this property characterizes the duality between scattering and bounded systems in the presence of local symmetries. Our findings are illustrated at hand of a numerical example with a potential consisting of two domains of local symmetry, each one comprised of Dirac ? barriers.