SUSY transformations with complex factorization constants. Application to spectral singularities

TL;DR

This paper explores SUSY transformations with complex constants in non-Hermitian quantum systems, introducing a regularization method for spectral singularities and analyzing eigenfunction properties at singular points.

Contribution

It develops a new regularization procedure for spectral singularities in non-Hermitian SUSY partner Hamiltonians and analyzes eigenfunction behavior at these singularities.

Findings

Regularization method for spectral singularities in non-Hermitian Hamiltonians

Eigenfunctions have zero binorm at spectral singularities

SUSY transformations with complex constants applied to spectral singularities

Abstract

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to Hamiltonians possessing spectral singularities which are non-Hermitian SUSY partners of selfadjoint operators. A new regularization procedure for the resolution of the identity operator in terms of continuous biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed. It is also shown that the continuous spectrum eigenfunction has zero binorm (in the sense of distributions) at the singular point.