Spectral singularities for Non-Hermitian one-dimensional Hamiltonians: puzzles with resolution of identity

TL;DR

This paper investigates the completeness of eigenfunctions for non-Hermitian Hamiltonians with spectral singularities, providing correct resolutions of identity for different potentials and clarifying the role of spectral singularities in physical state representations.

Contribution

It offers a detailed analysis of spectral singularities' impact on completeness and constructs proper resolutions of identity for various potential classes.

Findings

Spectral singularities do not obstruct completeness when properly accounted for.

Resolutions of identity depend on the class of functions used for physical states.

The study clarifies the role of spectral singularities in non-Hermitian quantum mechanics.

Abstract

We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.