Rigged Hilbert Space Approach for Non-Hermite Systems with Positive Definite Metric

TL;DR

This paper develops a rigged Hilbert space framework with positive-definite metric for non-Hermitian quantum systems, enabling spectral analysis and transformation theory, with applications to parity-time symmetric systems.

Contribution

It introduces a positive-definite metric rigged Hilbert space approach for non-Hermitian quantum systems, extending spectral and transformation analysis methods.

Findings

Spectral expansions for non-Hermitian operators established

Bi-orthogonal systems constructed within the framework

Application demonstrated for parity-time symmetric systems

Abstract

We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for a non-Hermite quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by positive-definite metric, is established. With the aid of the nuclear spectral theorem for the obtained rigged Hilbert space, spectral expansions are shown for the bra-kets by the generalized eigenvectors of a quasi-Hermite operator. The spectral expansions are utilized to endow the complete bi-orthogonal system and the transformation theory between the Hermite and non-Hermite systems. As an example of application, we show a specific description of our rigged Hilbert space treatment for some parity-time symmetrical quantum systems.