Semiclassical analysis of a quasi-exactly solvable system: second harmonic generation

Abstract

The usual second harmonic generation effective Hamiltonian is shown to be equivalent to a one-dimensional Schrodinger operator with a sextic polynomial potential. This operator belongs to the class of quasi-exactly solvable models, for which a finite number of eigenvalues and eigenvectors can be determined exactly. A Jeffreys-Wentzel-Kramers-Brillouin analysis provides accurate asymptotic expansions for these eigenvalues in the limit of large unperturbed energies.