Algebraic equations for the exceptional eigenspectrum of the generalised Rabi model

TL;DR

This paper derives algebraic equations characterizing the exceptional eigenspectrum of the generalized Rabi model, linking wavefunctions to algebraic roots and analyzing degeneracies and crossing points.

Contribution

It introduces a novel algebraic approach to determine the exceptional spectrum of the generalized Rabi model, connecting wavefunctions to roots of algebraic equations and comparing with existing methods.

Findings

Explicit algebraic equations for the exceptional spectrum.

Connection between wavefunctions and roots of algebraic equations.

Analysis of degeneracies and crossing points in the spectrum.

Abstract

We obtain the exceptional part of the eigenspectrum of the generalised Rabi model, also known as the driven Rabi model, in terms of the roots of a set of algebraic equations. This approach provides a product form for the wavefunction components and allows an explicit connection with recent results obtained for the wavefunction in terms of truncated confluent Heun functions. Other approaches are also compared. For particular parameter values the exceptional part of the eigenspectrum consists of doubly degenerate crossing points. We give a proof for the number of roots of the constraint polynomials and discuss the number of crossing points.