Construction of a new three boson non-hermitian Hamiltonian associated to deformed Higgs algebra: real eigenvalues and Partial PT-symmetry

TL;DR

This paper constructs a new non-Hermitian three-boson Hamiltonian related to a deformed Higgs algebra, demonstrating real eigenvalues and partial PT-symmetry, with the deformation parameter significantly affecting the eigenfunctions.

Contribution

It introduces a novel three-boson non-Hermitian Hamiltonian associated with a deformed Higgs algebra and explores its real eigenvalues and partial PT-symmetry properties.

Findings

Hamiltonian has real eigenvalues despite being non-Hermitian

Eigenstates exhibit symmetry-induced orthogonality

Deformation parameter influences eigenfunctions significantly

Abstract

A $\gamma$-deformed version of $\mathfrak{su}(2)$ algebra has been obtained from a bi-orthogonal system of vectors in $\bf{C^2}$. Fusion of Jordan-Schwinger realization of complexified $\mathfrak{su}(2)$ with Dyson-Maleev representation gives a 3-boson realization of Higgs algebra of cubic polynomial type. The non-hermitian Hamiltonian thus obtained is found to have real eigenvalues and eigen states with symmetry induced orthogonality. The notion of partial ${\mathcal {PT}}$-symmetry (henceforth $\partial_{\mathcal { PT}}$) has been introduced as a characteristic feature of these multi-boson realizations. The Hamiltonian along with its eigenstates have been studied in the light of $\partial_{\mathcal { PT}}$-symmetry. The possibility of $\partial_{\mathcal { PT}}$-symmetry breaking is also discussed. The deformation parameter $\gamma$ plays a crucial role in the entire formulation and non-trivially modifies the eigenfunctions under consideration.