  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.
