Quasi-Hermitian quantum mechanics and a new class of user-friendly matrix Hamiltonians

TL;DR

This paper introduces a new family of quasi-Hermitian Hamiltonians in quantum mechanics that simplifies the calculation of the associated metric operators, making the non-Hermitian formulation more user-friendly.

Contribution

The authors present a specific class of matrix Hamiltonians with explicitly obtainable metrics, enhancing practical usability of quasi-Hermitian quantum mechanics.

Findings

Derived closed-form expressions for the metric operators.

Identified a new class of matrix Hamiltonians with simplified structure.

Facilitated easier implementation of quasi-Hermitian models.

Abstract

In the conventional Schr\"{o}dinger's formulation of quantum mechanics the unitary evolution of a state $\psi$ is controlled, in Hilbert space ${\cal L}$, by a Hamiltonian $\mathfrak{h}$ which must be self-adjoint. In the recent, ``quasi-Hermitian'' reformulation of the theory one replaces $\mathfrak{h}$ by its isospectral but non-Hermitian avatar $H = \Omega^{-1}\mathfrak{h}\Omega$ with $\Omega^\dagger\Omega = \Theta \neq I$. Although acting in another, manifestly unphysical Hilbert space ${\cal H}$, the amended Hamiltonian $H \neq H^\dagger$ can be perceived as self-adjoint with respect to the amended inner-product metric $\Theta$. In our paper motivated by a generic technical ``user-unfriendliness'' of the non-Hermiticity of $H$ we introduce and describe a specific new family of Hamiltonians $H$ for which the metrics $\Theta$ become available in closed form.