Hermitian Hamiltonian equivalent to a given non-Hermitian one. Manifestation of spectral singularity

TL;DR

This paper investigates a simple non-Hermitian Hamiltonian with spectral singularity, showing its Hermitian equivalent has a supersymmetric structure, and spectral singularities manifest as resonances in scattering cross sections.

Contribution

It demonstrates the supersymmetric structure of the $ ext{eta}$ operator and explicitly relates spectral singularities to scattering resonances in the Hermitian equivalent.

Findings

Spectral singularity appears as a resonance in the scattering cross section.

Hermitian equivalent Hamiltonian becomes undetermined at the spectral singularity.

The $ ext{eta}$ operator has a supersymmetric second order differential structure.

Abstract

One of the simplest non-Hermitian Hamiltonians first proposed by Schwartz (1960 {\it Commun. Pure Appl. Math.} \tb{13} 609) which may possess a spectral singularity is analyzed from the point of view of non-Hermitian generalization of quantum mechanics. It is shown that $\eta$ operator, being a second order differential operator, has supersymmetric structure. Asymptotic behavior of eigenfunctions of a Hermitian Hamiltonian equivalent to the given non-Hermitian one is found. As a result the corresponding scattering matrix and cross section are given explicitly. It is demonstrated that the possible presence of the spectral singularity in the spectrum of the non-Hermitian Hamiltonian may be detected as a resonance in the scattering cross section of its Hermitian counterpart. Nevertheless, just at the singular point the equivalent Hermitian Hamiltonian becomes undetermined.