The minimally anisotropic metric operator in quasi-Hermitian quantum mechanics

TL;DR

This paper introduces a method to select a unique, minimally anisotropic inner product in quasi-Hermitian quantum mechanics, ensuring an operator becomes self-adjoint with minimal deviation from the original inner product.

Contribution

It proposes a novel variational approach to determine the optimal inner product, providing existence, uniqueness, and explicit equations for the minimally anisotropic metric.

Findings

Existence and uniqueness of the minimally anisotropic metric are established.

Derived Euler-Lagrange equations characterize the optimal inner product.

Examples show the optimal metric can differ from traditional choices.

Abstract

We propose a unique way how to choose a new inner product in a Hilbert space with respect to which an originally non-self-adjoint operator similar to a self-adjoint operator becomes self-adjoint. Our construction is based on minimising a 'Hilbert-Schmidt distance' to the original inner product among the entire class of admissible inner products. We prove that either the minimiser exists and is unique, or it does not exist at all. In the former case we derive a system of Euler-Lagrange equations by which the optimal inner product is determined. A sufficient condition for the existence of the unique minimally anisotropic metric is obtained. The abstract results are supplied by examples in which the optimal inner product does not coincide with the most popular choice fixed through a charge-like symmetry.