Resolutions of Identity for Some Non-Hermitian Hamiltonians. I. Exceptional Point in Continuous Spectrum

TL;DR

This paper develops resolutions of identity for specific non-Hermitian Hamiltonians with continuous spectra, focusing on exceptional points on or inside the spectrum boundary, and explores their spectral properties and associated functions.

Contribution

It introduces new resolutions of identity for non-Hermitian Hamiltonians with exceptional points, analyzing their properties and implications for spectral theory.

Findings

Some associated functions are normalizable at the exceptional point.

Bounded associated functions at the exceptional point may not belong to the physical space.

Spectral properties of SUSY partner Hamiltonians are examined.

Abstract

Resolutions of identity for certain non-Hermitian Hamiltonians constructed from biorthogonal sets of their eigen- and associated functions are given for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration possess the continuous spectrum and the following peculiarities are investigated: (1) the case when there is an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum; (2) the case when there is an exceptional point situated inside of continuous spectrum. The reductions of the derived resolutions of identity under narrowing of the classes of employed test functions are revealed. It is shown that in the case (1) some of associated functions included into the resolution of identity are normalizable and some of them may be not and in the case (2) the bounded associated function corresponding to the exceptional point does not belong to the physical state space. Spectral properties of a SUSY partner Hamiltonian for the Hamiltonian with an exceptional point are examined.