Tavis-Cummings models and their quasi-exactly solvable Schrödinger Hamiltonians

We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrodinger equations. The connection between them is established 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 Schrodinger potentials the quarkonium and the sextic oscillator appear.