Difference analogs of the harmonic oscillator

Solution of the Cauchy problem for a time-dependent Schrödinger equation

“…and the normalized wave functions are given by 9) where Y Kν (Ω) are the spherical harmonics associated with the tree T and the radial functions R N K (r) are defined by (2.7). The wave functions of the one-dimensional harmonic oscillator…”

“…We shall discuss this model and its generalization in the next two sections. Propagators for the difference models of the simple harmonic oscillator [9] can be derived in a similar fashion. The details are left to the reader.…”

Recurrence relations for discrete hypergeometric functions

“…Several of such models are solved using the theory of the classical discrete polynomials [22]. Important instances of such systems are the discrete oscillators of Charlier [5], Kravchuk oscillators [6,8,10,12,14] and Meixner oscillators [5] that are related to the polynomials of Charlier, Kravchuk and Meixner, respectively, and the finite radial oscillator [9,11] related with the Hahn polynomials. For applications it is important to have recurrence relations for the discrete wave function of such systems.…”

Four Lectures: Discretization and Integrability. Discrete Spectral Symmetries

“…Let us present here two interesting examples of the discrete 1D operators discussed from the algebraic point of view in the works [7,6] and also in [2],Appendix 2 from the viewpoint of the spectral theory.…”

“…Recently we developed a theory based on the discrete symmetries of the continuous and discrete 1D and 2D Schrodinger operators (see [2,3]). Some results for the 1D Schrodinger operators were obtained in the works [4,5,6,7].…”