Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs

TL;DR

This paper provides rigorous proofs for resolutions of identity related to non-Hermitian Hamiltonians with continuous spectra, focusing on cases involving exceptional points on or inside the spectrum boundary.

Contribution

It offers the first rigorous proofs for resolutions of identity for non-Hermitian Hamiltonians with exceptional points on the spectrum boundary or inside the spectrum.

Findings

Resolutions of identity are established for Hamiltonians with exceptional points.

Proofs cover cases with exceptional points on the boundary and inside the spectrum.

The work extends the mathematical foundation for non-Hermitian quantum mechanics.

Abstract

This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.