On a Relation among Bi-orthogonal system, Quadratic Non-Hermitian Boson operators with real spectrum and Partial PT symmetry in Fock Space

TL;DR

This paper explores partial PT symmetry in non-Hermitian quadratic boson operators within Fock space, analyzing eigenvalue reality and symmetry breaking through a deformation parameter.

Contribution

It introduces the concept of partial PT symmetry for non-Hermitian quadratic boson operators and studies its implications in Fock space as a Reproducing Kernel Hilbert Space.

Findings

Eigenvalues can be real under certain symmetry conditions.

Partial PT symmetry influences the reality of eigenvalues.

Symmetry breaking occurs depending on the deformation parameter.

Abstract

A new kind of symmetry called partial PT symmetry has been considered for non-hermitian quadratic boson operators obtained from a bi-orthogonal set of vectors in C2. The symmetry behaviour has been understood in Fock space considered as a Reproducing Kernel Hilbert Space(RKHS). The reality of eigenvalues and its connection to the possibility of the aforesaid symmetry (and symmetry breaking) are studied in terms of a deformation parameter responsible for nonhermiticity