Tavis-Cummings models and their quasi-exactly solvable Schr\"odinger Hamiltonians

TL;DR

This paper explores the connection between Tavis-Cummings models in quantum optics and quasi-exactly solvable Schrödinger equations, revealing new insights into their mathematical structure and specific potentials like quarkonium.

Contribution

It establishes a detailed relationship between Tavis-Cummings Hamiltonians and quasi-exactly solvable Schrödinger equations via the biconfluent Heun equation, identifying specific potentials with known solutions.

Findings

Each invariant subspace corresponds to potentials with known solutions.

Identifies quarkonium and sextic oscillator as relevant potentials.

Connects quantum optics models with solvable Schrödinger equations.

Abstract

We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schr\"odinger equations. The connection between them is stablished through the biconfluent Heun equation. We found that each invariant $n$-dimensional subspace of Tavis-Cummings Hamiltonian corresponds either to $n$ potentials, each with one known solution, or to one potential with $n$-known solutions. Among these Schr\"odinger potentials appear the quarkonium and the sextic oscillator.