Calculation of the metric in the Hilbert space of a PT-symmetric model via the spectral theorem

TL;DR

This paper presents an alternative, more elegant formula for the metric operator in a PT-symmetric quantum model, using the spectral theorem to deepen understanding of the model's structure.

Contribution

It introduces a novel formula for the metric operator in a PT-symmetric model, leveraging the spectral theorem for self-adjoint operators.

Findings

New formula for the metric operator derived

Spectral theorem applied to PT-symmetric models

Enhanced understanding of model's mathematical structure

Abstract

In a previous paper (arXiv:math-ph/0604055) we introduced a very simple PT-symmetric non-Hermitian Hamiltonian with real spectrum and derived a closed formula for the metric operator relating the problem to a Hermitian one. In this note we propose an alternative formula for the metric operator, which we believe is more elegant and whose construction -- based on a backward use of the spectral theorem for self-adjoint operators -- provides new insights into the nature of the model.