SPECIAL POLYNOMIALS IN PROBLEMS OF QUANTUM OPTICS

For a general Raman-type model describing the decay of a pumping mode into p components special polynomials are introduced, Asymptotic properties of those polynomials are discussed when the number of excitations tends to infinity. In the investigation of interaction between radiation and media, a number of mathematical complexities arises. The simplest example is the modeling of radiation scattering by excitations of a crystal lattice. Some of the contradictions are caused by the peculiarities of the second quantization method, which is used effectively for quadratic Hamiltonians only. Another problem is the determination of the energy states of the investigated media, which is a problem of eigenvalues. Mathematically we are to examine some system of polynomials connecting each other by a recurrency, which is analogous to the classical case. In this paper we consider the mathematical problems arising therein.