We study in detail the relationship between the Tavis-Cummings Hamiltonian of quantum optics and a family of quasi-exactly solvable Schrödinger 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 Schrödinger potentials appear the quarkonium and the sextic oscillator.PACS: 42.65.Ky, 02.20.Sv, 03.65.Fd, 02.30.Gp IntroductionThe main objective of this paper is to show a direct method to connect quantum optics Hamiltonians with quasi-exactly solvable (QES) one-dimensional Schrödinger equations by considering the case of a trilinear Hamiltonian. In principle, these two topics are quite far away since optical Hamiltonians make use of operators corresponding to the radiation modes or to the interacting atoms with a number of allowed transitions, while Schrödinger equations describe a (one dimensional, in this case) particle under an external potential. However, the relation between the solutions of quantum optical Hamiltonians and Schrödinger wavefunctions has been already considered in some references (see [1][2][3]). Our purpose in this work is to illustrate this connection in a clear and explicit way, including all the relevant details.The quantum optical Hamiltonians we consider are modifications of the Jaynes-Cummings model [4] in the rotating wave approximation (or RWA) describing the interaction of a onemode quantum radiation field with a two-level atom. Here, we will specially be concerned with the Dicke or Tavis-Cummings (TC) Hamiltonians which take into account the interaction of radiation with a population of two-level atoms [5] (other cases such as three-level atoms can also be considered, but it will not be done here). With the help of the Schwinger representation we

We study in detail the behavior of the energy spectrum for the second harmonic generation (SHG) and a family of corresponding quasi-exactly solvable Schrödinger potentials labeled by a real parameter b. The eigenvalues of this system are obtained by the polynomial deformation of the Lie algebra sl(2, R) representation space. We have found the bi-confluent Heun equation (BHE) corresponding to this system in a differential realization approach, by making use of the symmetries. By means of a b-transformation from this second-order equation to a Schrödinger one, we have found a family of quasi-exactly solvable potentials. For each invariant n-dimensional subspace of the second harmonic generation, there are either n potentials, each with one known solution, or one potential with n-known solutions. Well-known potentials like a sextic oscillator or that of a quantum dot appear among them.

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