# Theory of response to perturbations in non-Hermitian systems using five-Hilbert-space reformulation of unitary quantum mechanics

## TL;DR

This paper develops a five-Hilbert-space framework to analyze how non-Hermitian, non-unitary quantum systems respond to perturbations, extending the understanding of stability and dynamics in PT-symmetric and relativistic quantum mechanics.

## Contribution

It introduces a novel five-Hilbert-space reformulation to handle perturbations in non-Hermitian quantum systems, improving the analysis of their stability and response.

## Key findings

- Reformulation clarifies the role of inner-product metrics in non-Hermitian systems.
- Perturbation effects are systematically analyzed within the new framework.
- Implications for stability analysis in PT-symmetric and relativistic quantum mechanics are discussed.

## Abstract

In conventional Schr\"{o}dinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=\Omega\,\mathfrak{h}\,\Omega \neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\cal L}$. The situation encountered, typically, in ${\cal PT}-$symmetric or relativistic quantum mechanics is shown more complicated after an inclusion of perturbations. The formulation and solution of the problem are presented. Some of the consequences relevant, e.g., in the analysis of stability are discussed.