Scattering cross section resonance originating from a spectral singularity

TL;DR

This paper explores how spectral singularities in non-Hermitian Hamiltonians manifest as resonances in the scattering cross section of their Hermitian counterparts, enabling potential experimental detection of such singularities.

Contribution

It demonstrates the connection between spectral singularities and resonant scattering in Hermitian Hamiltonians using supersymmetric quantum mechanics techniques.

Findings

Spectral singularities cause pronounced resonances in Hermitian scattering cross sections.

Hermitian and non-Hermitian scattering matrices are related through phase factors.

Resonance behavior is explained by the square root of the Breit-Wigner matrix.

Abstract

Using techniques of supersymmetric quantum mechanics, scattering properties of Hermitian Hamiltonians, which are related to non-Hermitian ones by similarity transformations, are studied. We have found that the scattering matrix of the Hermitian Hamiltonian coincides with the phase factor of the non-unitary scattering matrix of the non-Hermitian Hamiltonian. The possible presence of a spectral singularity in a non-Hermitian Hamiltonian translates into a pronounced resonance in the scattering cross section of its Hermitian counterpart. This opens a way for detecting spectral singularities in scattering experiments; although a singular point is inaccessible for the Hermitian Hamiltonian, the Hamiltonian "feels" the presence of the singularity if it is "close enough". We also show that cross sections of the non-Hermitian Hamiltonian do not exhibit any resonance behavior and explain the resonance behavior of the Hermitian Hamiltonian cross section by the fact that corresponding scattering matrix, up to a background scattering matrix, is a square root of the Breit-Wigner scattering matrix.