Rigged Liouville space formulation for quasi-Hermitian Liouville operators

TL;DR

This paper develops a rigorous super bra-ket formalism for quasi-Hermitian Liouvillian operators using rigged Liouville space, enabling spectral analysis of both Hermitian and non-Hermitian systems.

Contribution

It introduces a novel rigged Liouville space framework that unifies spectral decomposition of Hermitian and quasi-Hermitian Liouville operators.

Findings

Constructed a rigorous foundation for super bra-ket formalism in RHS.

Demonstrated spectral decomposition for Hermitian and non-Hermitian harmonic oscillators.

Showed differences in spectral forms between Hermitian and non-Hermitian cases.

Abstract

We discuss a super bra-ket formalism for quasi-Hermitian Liouvillian operators within the framework of rigged Hilbert spaces (RHS). An RHS in terms of the Liouville space, referred to as a rigged Liouville space (RLS), is reconstructed by exploiting the mathematical fact that the space of Hilbert-Schmidt operators is unitarily equivalent to the tensor product of Hilbert spaces. The obtained RLS endows a rigorous foundation of the construction for the super bra-ket and for the spectral decompositions of both Hermitian and quasi-Hermitian Liouville operators, which are characterized by the generalized eigenvectors in the dual spaces. Furthermore, within this framework, the non-Hermitian Liouvillian operator and its adjoint can be constructed symmetrically, with their symmetric structure preserved. As an application of our RLS methodology, we examine the Liouville operators corresponding to Hermitian and non-Hermitian harmonic oscillators and elucidate the differences between their spectral decomposition forms.