M. Simon and  S. K. Suslov, Expansion of analytic functions in q-orthogonal polynomials, The Ramanujan Journal,  vol. 19, pp. 281-303, 2009.

http://www.springerlink.com/content/93468l2q34218753/

Relativistic Kramers-Pasternack Recurrence Relations
Author: Sergei K. Suslov

http://arxiv.org/abs/0908.3021

Abstract: Recently we have evaluated the matrix elements $<Or^{p}>$,$ where $O$ $={1,\beta, i\mathbf{\alpha n}\beta} $ are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions $_{3}F_{2}(1) $ for all suitable powers and established two sets of Pasternack-type matrix identities for these integrals. The corresponding Kramers--Pasternack type three-term vector recurrence relations are studied here.

A Mechanism for Stabilization of Dynamics in Nonlinear Systems with Different Time Scales
Authors: Raquel M. Lopez, Erika T. Camacho, and Sergei K. Suslov

http://arxiv.org/abs/0907.1609

Abstract: There are many natural, physical, and biological systems that exhibit multiple time scales. For example, the dynamics of a population of ticks can be described in continuous time during their individual life cycle yet discrete time is used to describe the generation of offspring. These characteristics cause the population levels to be reset periodically. A similar phenomenon can be observed in a sociological college drinking model in which the population is reset by the incoming class each year, as described in the 2006 work of Camacho et al. With the latter as our motivation we analytically and numerically investigate the mechanism by which solutions in certain systems with this resetting conditions stabilize. We further utilize the sociological college drinking model as an analogue to analyze certain one-dimensional and two-dimensional nonlinear systems, as we attempt to generalize our results to higher dimensions.

Expectation Values $<r^p>$ for Harmonic Oscillator in $R^n$
Authors: Ricardo Cordero-Soto and Sergei K. Suslov

http://arxiv.org/abs/0908.0032

Abstract: We evaluate the matrix elements $<r^{p}>$ for the $n$ -dimensional harmonic oscillator in terms of the dual Hahn polynomials and derive the corresponding three term recurrence relation and the Pasternack-type reflection relation. A short review of similar results is also given.

Expectation Values in Relativistic Coulomb Problems
Authors: Sergei K. Suslov

Comments: published in J. Phys. B: At. Mol. Opt. Phys. September 2009: 
42 185003 (8pp)   doi: 10.1088/0953-4075/42/18/185003 

http://stacks.iop.org/0953-4075/42/185003

see also:

http://arxiv.org/abs/0906.3338

Abstract:  We evaluate the matrix elements <Or^{p}>, where O ={1, \beta, i\alpha n \beta} are the standard Dirac matrix operators for the relativistic Coulomb problems, in terms of the generalized hypergeometric functions_{3}F_{2} for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we derive two sets of Pasternack-type matrix identities for these integrals, when p->-p-1 and p->-p-3, respectively. Some applications to the theory of hydrogenlike relativistic systems are reviewed.

Models of Damped Oscillators in Quantum Mechanics
Authors: Ricardo Cordero-Soto, Erwin Suazo, Sergei K. Suslov

http://arxiv.org/abs/0905.0507

Abstract: We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration.

Comments: To appear in Journal of Physical Mathematics, Vol.1 (2009), Article ID S090603,16 pages, doi:10.4303/jpm/S090603

The time-dependent Schroedinger equation, Riccati equation and Airy functions
Authors: Nathan Lanfear, Sergei K. Suslov

http://arxiv.org/abs/0903.3608

Abstract: We construct the Green functions (or Feynman's propagators) for the Schroedinger equations of the form $i\psi_{t}+{1/4}\psi_{xx}\pm tx^{2}\psi =0$ in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schroedinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is stablished.
Comments: 28 pages, one figure

The Time Inversion for Modified Oscillators
Authors: Ricardo Cordero-Soto, Sergei K. Suslov

http://arxiv.org/abs/0808.3149

Abstract: We discuss a new completely integrable case of the time-dependent Schroedinger equation in $R^n$ with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is also found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. Particular solutions of the corresponding Schroedinger equations are also obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.
Comments: 33 pages, four figures, and two tables; to appear in Teoret.and Math. Phys.

The Riccati Differential Equation and a Diffusion-Type Equation
Authors: Erwin Suazo, Sergei K. Suslov, Jose M. Vega-Guzman

http://arxiv.org/abs/0807.4349

Abstract: We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equations with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding non-homogeneous equation is also found.
Comments: 12 pages, no figures

An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients
Authors: Erwin Suazo, Sergei Suslov

http://arxiv.org/abs/0805.0633

Abstract: We discuss an integral form of the Cauchy initial value problem for the nonlinear Schroedinger equation with variable coefficients. Some special and limiting cases are outlined.
Comments: 14 pages, no figures

Solution of the Cauchy problem for a time-dependent Schroedinger equation
J. Math. Phys. bf{49} (2008) \#7, 072102: pp. 1--27; published on line 9 July 2008.
Authors: Maria Meiler, Ricardo Cordero-Soto, and Sergei K. Suslov

http://link.aip.org/link/?JMP/49/072102.

Abstract: We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products. ©2008 American Institute of Physics

Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields
Lett. Math. Phys. bf{84} (2008)~\#2--3, 159-178.
Authors: Ricardo Cordero-Soto, Raquel M. Lopez, Erwin Suazo, and Sergei K. Suslov

http://www.springerlink.com/content/h523m0u4m378l266/fulltext.pdf

Abstract: We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schroeinger equation for a charged particle with a spin moving in a uniform magnetic field and a perpendicular electric field varying with time. The corresponding Green function (propagator) is given in terms of elementary functions and certain integrals of the fields with a characteristic function, which should be found as an analytic or numerical solution of the equation of motion for the classical oscillator with a time-dependent frequency. We discuss a particular solution of a related nonlinear Schroeinger equation and some special and limiting cases are outlined.

The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems
J. Math. Phys. 49, 012104 (2008); DOI:10.1063/1.2830804
Published 22 January 2008
Authors: Sergei K. Suslov and Benjamin Trey

http://link.aip.org/link/?JMAPAQ/49/012104/1

Abstract: We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic Coulomb problems in terms of the Hahn and Chebyshev polynomials of a discrete variable. A short review on special functions and solution of the Coulomb problems in quantum mechanics is given.
Comments: 51 pages, no figures

K. Ey, A. Ruffing, and S. K. Suslov, Method of separation of the variables for basic analogs of equations of mathematical physics, The Ramanujan Journal, Askey Special Issues (G. E. Andrews, G. Gasper, and S. K. Suslov, Coordinating Editors), Vol. 13, No. 1-3, pp. 407-447, 2007.

I was one of the coordinating editors of a major publication in the area of special functions and q-orthogonal polynomials, please see:

http://www.math.ufl.edu/~frank/ramanujan/vol13/issue1-3/toc.html

The Ramanujan Journal, Askey Special Issues, Volume 13, Number 1-3, June 2007, pp. 1-469; G. E. Andrews, G. Gasper, and S. K. Suslov, Coordinating Editors.

S. K. Suslov, An analog of the Cauchy – Hadamard formula for expansions in q-orthogonal polynomials, in: Theory and Applications of Special Functions  (M. E. H. Ismail and Erik Koelink, eds.), Springer Series Developments in Mathematics, Vol. 13, Springer – Verlag, 2005, pp. 443-460.

S. K. Suslov, An algebra of integral operators, Electronic Transactions on Numerical Analysis (ETNA), Vol. 27, pp. 140-155, 2007

http://etna.mcs.kent.edu/vol.27.2007/pp140-155.dir/pp140-155.html

R. M. Lopez and S. K. Suslov, The Cauchy problem for a forced harmonic oscillator

http://arxiv.org/abs/0707.1902