su(2) and su(1,1) operators within N=2 supersymmetry

Approximation Theory and Special Functions ATSF 2024 Conference - 8th Series, Ankara, Türkiye, 4 - 07 Eylül 2024

A class of non-Hermitian Hamiltonian operators with real spectra that are expressed using su(2)

and su(1, 1) generators is studied [1]. The Lie algebra su(2) is a fundamental concept in quantum

mechanics, particularly in the context of angular momentum and spin. It is associated with the

special unitary group SU(2), which is the group of 2×2 unitary matrices with determinant 1. The

Lie algebra su(2) consists of all 2×2 traceless Hermitian matrices. In this study, su(2) operators are

studied in the context of supersymmetric quantum mechanics. Moreover, in quantum mechanics,

the su(1, 1) Lie algebra plays a significant role, particularly in the context of systems with one-

dimensional radial symmetry, such as the hydrogen atom or the Morse potential. This Lie algebra

is relevant in these contexts because it describes the dynamical symmetries of these systems,

particularly those associated with angular momentum. The connection between su(1, 1) and su(2)

algebras arises in certain contexts, particularly in the study of symmetries and dynamical properties

of quantum systems. While the structures of the algebras are different, they often share similar

mathematical properties and can sometimes be related through mathematical transformations

or mappings. This connection has been explored in various areas of quantum mechanics and

mathematical physics. This study is extended to su(1, 1) systems as well. As a toy model, the

Swanson Hamiltonian [2] is studied. The construction of isospectral Hamiltonians is shown, and

exact solutions for the corresponding systems are obtained.