Department of Mechanics and Mathematics
Kharkov V. N. Karazin National University
Svoboda Square 4, Kharkov 61077, Ukraine
E-mail: aptm@kharkov.ua

Quasi-exactly solvable Bose Hamiltonians
by S. N. Dolya and O. B. Zaslavskii

Abstract:
We extend the notion of quasi-exactly solvable (QES) models from potential ones and differential equations to Bose systems. We consider Hamiltonians, polynomial with respect to Bose operators and obtain conditions under which algebraization of the part of the spectrum occurs. The basis functions look rather simple in the coherent state representation but include, in general, an infinite number of quasi-particles, corresponding to the original Bose operators. These functions are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labeling the coherent state representation. In some particular degenerate cases they turn (up to the power factor) into the trigonometric or hyperbolic functions, Bessel functions or combinations of the exponent and Hermit polynomials.