Systematics of quasi-Hermitian representations of non-Hermitian quantum models

TL;DR

This paper explores different methods of constructing physically consistent quantum models using non-Hermitian Hamiltonians, focusing on the systematic transition between various Hilbert space representations and their associated metrics.

Contribution

It introduces a unified framework for transitioning between different quasi-Hermitian representations, including hybrid approaches that modify both the metric and the Hamiltonian.

Findings

Multiple representations of non-Hermitian quantum models are systematically characterized.

A hybrid construction method for quantum models is proposed and exemplified.

The framework aids in selecting optimal representations for specific Hamiltonians.

Abstract

In the recently quickly developing context of quantum mechanics of unitary systems using a time-independent non-Hermitian Hamiltonian $H$ (having real spectrum and defined as acting in an unphysical but user-friendly Hilbert space ${\cal R}_N^{(0)}$), the present paper introduces and describes a set of constructive returns of the description to one of the correct and eligible physical Hilbert spaces ${\cal R}_0^{(j)}$. The superscript $j$ may run from $j=0$ to $j=N$. In the $j=0$ extreme of the theory the construction is currently well known and involves solely the inner product metric $\Theta=\Theta(H)$. The Hamiltonian $H$ itself remains unchanged. At $j=N$ the inner-product metric remains trivial and only the Hamiltonian must be Hermitized, $H \to \mathfrak{h} = \Omega\,H\,\Omega^{-1}=\mathfrak{h}^\dagger$. At the remaining superscripts $j=1,2,\ldots, N-1$, a new, hybrid form of the construction of a consistent quantum model is proposed, requiring a simultaneous amendment of both the metric and the Hamiltonian. In applications, one of these options is expected to be optimal for a given $H$ in a way illustrated by a schematic three-state example.