Rigged Hilbert Space formulation for quasi-Hermitian composite systems

TL;DR

This paper develops a rigged Hilbert space framework for quasi-Hermitian composite quantum systems, enabling spectral analysis and bra-ket formalism extension to dual spaces, with applications to non-Hermitian harmonic oscillators.

Contribution

It introduces a positive definite metric RHS for quasi-Hermitian systems and extends bra-ket formalism to dual spaces, advancing the mathematical foundation of non-Hermitian quantum mechanics.

Findings

Constructed a positive definite metric RHS for quasi-Hermitian systems

Extended bra-ket formalism to dual spaces in non-Hermitian context

Applied framework to non-Hermitian harmonic oscillator systems

Abstract

The discussion in this study delves into Dirac's bra-ket formalism for a quasi-Hermitian quantum composite system based on the rigged Hilbert space (RHS). We establish an RHS with a positive definite metric suitable for a quasi-Hermite composite system. The obtained RHS is utilized to construct the bra and ket vectors for the non-Hermite composite system and produce the spectral decomposition of the quasi-Hermitian operator. We show that the symmetric relations regarding quasi-Hermitian operators can be extended to dual spaces, and all descriptions obtained using the bra-ket formalism are completely developed in the dual spaces. Our methodology is applied to a non-Hermitian harmonic oscillator composed of conformal multi-dimensional many-body systems.