Continuum states in generalized Swanson models

TL;DR

This paper explores the continuum states of a non-Hermitian Swanson-type Hamiltonian and their relation to an equivalent Hermitian Schrödinger Hamiltonian, revealing insights into their spectral correspondence and state characteristics.

Contribution

It extends the spectral analysis of Swanson models by focusing on continuum states and their correspondence through similarity transformations.

Findings

Established the nature of continuum states in non-Hermitian Swanson models.

Analyzed the spectral correspondence between non-Hermitian and Hermitian Hamiltonians.

Provided insights into the physical implications of continuum states in these models.

Abstract

A one-to-one correspondence is known to exist between the spectra of the discrete states of the non Hermitian Swanson-type Hamiltonian $ H = {\cal{A}}^{\dagger} {\cal{A}} + \alpha {\cal{A}} ^2 + \beta {\cal{A}}^{\dagger 2} $, ($\alpha \neq \beta $), and an equivalent Hermitian Schr\"{o}dinger Hamiltonian $h$, the two Hamiltonians being related through a similarity transformation. In this work we consider the continuum states of $h$, and examine the nature of the corresponding states of $H$.