Algebraic treatment of non-Hermitian quadratic Hamiltonians

TL;DR

This paper extends an algebraic method to analyze non-Hermitian quadratic Hamiltonians, allowing eigenvalue determination via matrix eigenvalues while considering symmetries, simplifying previous approaches.

Contribution

It introduces a generalized algebraic framework for non-Hermitian Hamiltonians, incorporating symmetries and avoiding direct Schrödinger equation solutions.

Findings

Eigenvalues obtained from matrix representations match known results.

The method effectively accounts for unitary and antiunitary symmetries.

Application to various quadratic Hamiltonians demonstrates versatility.

Abstract

We generalize a recently proposed algebraic method in order to treat non-Hermitian Hamiltonians. The approach is applied to several quadratic Hamiltonians studied earlier by other authors. Instead of solving the Schr\"odinger equation we simply obtain the eigenvalues of a suitable matrix representation of the operator. We take into account the existence of unitary and antiunitary symmetries in the quantum-mechanical problem.