A New Boson realization of Fusion Polynomial Algebras in Non-Hermitian Quantum Mechanics : $\gamma$-deformed $su(2)$ generators, Partial $\mathcal{P}\mathcal{T}$-symmetry and Higgs algebra

Abstract

A $\gamma$-deformed version of $su(2)$ algebra with non-hermitian generators has been obtained from a bi-orthogonal system of vectors in ${\mathbf{C}}^2$. The related Jordan-Schwinger(J-S) map is combined with boson algebras to obtain a hierarchy of fusion polynomial algebras. This makes possible the construction of Higgs algebra of cubic polynomial type. Finally the notion of partial $\mathcal{P}\mathcal{T}$ symmetry has been introduced as characteristic feature of some operators as well as their eigenfunctions. The possibility of partial $\mathcal{P}\mathcal{T}$-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.

References in corpus (10)

• Visualization of Branch Points in PT-Symmetric Waveguides
• PT-Symmetric Phonon Laser
• A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points
• Bifurcation diagram and pattern formation in superconducting wires with electric currents
• Consistency of PT-symmetric quantum mechanics
• Coupled Oscillator Systems Having Partial PT Symmetry
• Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras
• Tilted optical lattices with defects as realizations of PT symmetry in Bose-Einstein condensates
• Spectrum and normal modes of non-hermitian quadratic boson operators
• Truncation of the reflection algebra and the Hahn algebra

Cited by in corpus (1)

• Instantons and transseries of the Mathieu potential deformed by a $\mathcal{P}\mathcal{T}$-symmetry parameter