Quasi-exactly solvable quartic Bose Hamiltonians

Abstract

We consider Hamiltonians, which are even polynomials of the forth order with the respect to Bose operators. We find subspaces, preserved by the action of Hamiltonian These subspaces, being finite-dimensional, include, nonetheless, states with an infinite number of quasi-particles, corresponding to the original Bose operators. The basis functions look rather simple in the coherent state representation and are expressed in terms of the degenerate hypergeometric function with respect to the complex variable labeling the 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. We find explicitly the relationship between coefficients at different powers of Bose operators that ensure quasi-exact solvability of Hamiltonian.

Quasi-exactly solvable models based on special functions

“…It is worth stressing that in [17] the eigenfunctions were constructed on the basis of elementary function, whereas the solutions (47, 48) are constructed on the basis of functions Table1. Values of parameters Hamiltonian (44,45), for different dim(R To the best of my knowledge, the only previously known example of QES related to special functions was found in [19].where these function appeared "by chance" for a particular problem connected with quartic Bose Hamiltonians. Now, we developed a systematic approach of QES-extension that enables us to generate new QES operators based on special functions.…”

Solutions of Two-Mode Bosonic and Transformed Hamiltonians

“…The single boson realization of the SU(1, 1) algebra has been studied in [39,40]. In this work we follow a different strategy to obtain the condition for QES of the two-bosonic systems.…”

Quasi-exactly solvable models in nonlinear optics

“…(Later, the same authors [9] used this method to study the problem of third harmonic generation and pointed out that it works in general for the n-th harmonic generation.) Two recent papers by Dolya and Zaslavskii [10,11] proceed in a related but different direction: they study models with a single degree of freedom whose Hamiltonians are even polynomials in the creation and annihilation operators, and show that under certain conditions these Hamiltonians also lead to a QES system.…”

“…Two recent papers by Dolya and Zaslavskii [10,11] proceed in a related but different direction: they study models with a single degree of freedom whose Hamiltonians are even polynomials in the creation and annihilation operators, and show that under certain conditions these Hamiltonians also lead to a QES system.…”