sl(2, C) as a complex Lie algebra and the associated non-Hermitian Hamiltonians with real eigenvalues

TL;DR

This paper extends the group theoretical formalism to non-Hermitian Hamiltonians with real spectra by complexifying the algebra so(2,1), resulting in new PT-symmetric and non-PT-symmetric models.

Contribution

It introduces the complex Lie algebra sl(2,C) as a framework for analyzing non-Hermitian Hamiltonians with real eigenvalues, expanding the potential algebra approach.

Findings

Development of a complex algebraic formalism for non-Hermitian Hamiltonians

Construction of new PT-symmetric and non-PT-symmetric Hamiltonians

Demonstration of real eigenvalues in the extended algebraic framework

Abstract

The powerful group theoretical formalism of potential algebras is extended to non-Hermitian Hamiltonians with real eigenvalues by complexifying so(2,1), thereby getting the complex algebra sl(2,\C) or $A_1$. This leads to new types of both PT-symmetric and non-PT-symmetric Hamiltonians.