Exceptional points of the eigenvalues of parameter-dependent Hamiltonian operators

TL;DR

This paper develops a method to efficiently find exceptional points of eigenvalues in parameter-dependent Hamiltonians by using the discriminant of the secular determinant, simplifying the problem to polynomial root-finding.

Contribution

It introduces a discriminant-based approach to locate exceptional points in various physical Hamiltonian models, streamlining the analysis process.

Findings

The method simplifies the calculation of exceptional points.

Application to diverse models demonstrates versatility.

Roots of polynomial functions identify critical parameter values.

Abstract

We calculate the exceptional points of the eigenvalues of several parameter-dependent Hamiltonian operators of mathematical and physical interest. We show that the calculation is greatly facilitated by the application of the discriminant to the secular determinant. In this way the problem reduces to finding the roots of a polynomial function of just one variable, the parameter in the Hamiltonian operator. As illustrative examples we consider a particle in a one-dimensional box with a polynomial potential, the periodic Mathieu equation, the Stark effect in a polar rigid rotor and in a polar symmetric top.