Alexei Zhedanov

Generalized little -Jacobi polynomials as eigensolutions of higher-order -difference operators

We consider the polynomials pn(x; a, b;M) obtained from the little q-Jacobi polynomials pn(x; a, b) by inserting a discrete mass M at x = 0 in the orthogonality measure. We show that for a = q, j = 0, 1, 2, . . . the polynomials pn(x; a, b;M) are eigensolutions of a linear q-difference operator of order 2j+4 with polynomial coefficients. This provides a q-analog of results recently obtained for the Krall polynomials.

Two-dimensional Krall-Sheffer polynomials and integrable systems

Two-dimensional Krall-Sheffer polynomials are analogues of the classical orthogonal polynomial. They are eigenfunctions of second-order linear partial differential operators and moreover satisfy orthogonality conditions. We show that all Krall-Sheffer polynomials are connected with two-dimensional superintegrable systems on spaces with constant curvature.

d-Orthogonal Charlier Polynomials and the Weyl Algebra

It is shown that d-orthogonal Charlier polynomials arise as matrix elements of non unitary automorphisms of the Weyl algebra. The structural formulas that these polynomials obey are derived from this algebraic setting.

SPECTRAL ANALYSIS OF Q-OSCILLATOR WITH GENERAL BILINEAR INTERACTION

Spectra of the most general Hermitian Hamiltonian which is bilinear in the creation and annihilation operators of a q-harmonic oscillator are investigated with the help of the factorization method. It is shown that there are two factorization schemes leading to discrete spectra of complicated forms. For q-oscillator models with continuous spectrum the Hamiltonian may have normalizable eigenstates of infinite multiplicity. Existence of the continuous spectrum in the interacting system is also discussed.