Dirichlet and Neumann Problems for String Equation, Poncelet Problem and Pell-Abel Equation

We consider conditions for uniqueness of the solution of the Dirichlet or the Neumann problem for 2-dimensional wave equation inside of bi-quadratic algebraic curve. We show that the solution is non-trivial if and only if corresponding Poncelet problem for two conics associated with the curve has periodic trajectory and if and only if corresponding Pell-Abel equation has a solution

The generalized MIC-Kepler system

This paper deals with dynamical system that generalizes the MIC-Kepler system. It is shown that the Schr\"{o}dinger equation for this generalized MIC-Kepler system can be separated in spherical and parabolic coordinates. The spectral problem in spherical and parabolic coordinates is solved.Comment: 8 page

Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of $q$-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

Novel Bound States Treatment of the Two Dimensional Schrodinger Equation with Pseudocentral Plus Multiparameter Noncentral Potential

By converting the rectangular basis potential V(x,y) into the form as V(r)+V(r, phi) described by the pseudo central plus noncentral potential, particular solutions of the two dimensional Schrodinger equation in plane-polar coordinates have been carried out through the analytic approaching technique of the Nikiforov and Uvarov (NUT). Both the exact bound state energy spectra and the corresponding bound state wavefunctions of the complete system are determined explicitly and in closed forms. Our presented results are identical to those of the previous works and they may also be useful for investigation and analysis of structural characteristics in a variety of quantum systemsComment: Published, 16 page

An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and elliptic)

Elliptic 6j-symbols first appeared in connection with solvable models of statistical mechanics. They include many interesting limit cases, such as quantum 6j-symbols (or q-Racah polynomials) and Wilson’s biorthogonal 10W9 functions. We give an elementary construction of elliptic 6j-symbols, which immediately implies several of their main properties. As a consequence, we obtain a new algebraic interpretation of elliptic 6j-symbols in terms of Sklyanin algebra representations