Trace formula and spectral Riemann surfaces for a class of tri-diagonal matrices

TL;DR

This paper investigates the spectral properties of a class of tri-diagonal matrices from the Jaynes--Cummings model, providing eigenvalue asymptotics, a trace formula, and proving the irreducibility of the spectral Riemann surface.

Contribution

It introduces new asymptotic formulas, a trace formula, and establishes the irreducibility of the spectral Riemann surface for these matrices.

Findings

Eigenvalues asymptotics derived

Trace formula proven

Spectral Riemann Surface shown to be irreducible

Abstract

For tri-diagonal matrices arising in the simplified Jaynes--Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible.