# Recent preprints&papers

by Sergei K. Suslov

School of Mathematics and Statistics Sciences&Mathematical, Computational and Modeling Sciences Center
Arizona State University
Tempe, AZ 85287-1804
USA

Title:  On the Harmonic Oscillator Group
Authors: Raquel M. Lopez, Sergei K. Suslov and Jose Vega-Guzman
Comments: 11 pages, no figures; for some Mathematica proofs and animations, please see BerrySummary HarmonicOscillatorGroup.nb
(For Mathematica proofs of Lemmas 1-3, please see: Koutschan.nb )
Abstract:
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a viewpoint of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schroedinger group of the free particle.
http://arxiv.org/abs/1111.5569

Title:  Dynamical Invariants and Berry's Phase for Generalized Driven Harmonic Oscillators
Authors: Barbara Sanborn, Sergei K. Suslov and Luc Vinet
Comments: 10 pages, no figures; published in Journal of Russian Laser Research, Volume 32, Issue 5 (2011), Pages 486-494; http://www.springerlink.com/openurl.asp?genre=article&id=doi:10.1007/s10946-011-9238-7
For some Mathematica details, please see BerrySummary
Abstract: We present quadratic dynamical invariant and evaluate Berry's phase for the time-dependent Schroedinger equation with the most general variable quadratic Hamiltonian.
http://arxiv.org/abs/1108.5144

Title: Exact Wave Functions for Generalized Harmonic Oscillators
Authors:  Nathan Lanfear, Raquel M. Lopez and Sergei K. Suslov
Comments: 10 pages, no figures; published in Journal of Russian Laser Research, Volume 32, Issue 4 (2011), Pages 352-361: for Mathematica proofs of Lemmas 1-3, please see: Koutschan.nb

Abstract: We construct exact wave functions for the most general variable quadratic Hamiltonians in terms of solutions of certain Ermakov and Riccati-type systems.
http://arxiv.org/abs/1102.5119

Title: The Riccati System and a Diffusion-Type Equation
Authors: Erwin Suazo, Sergei K. Suslov, and Jose M. Vega-Guzman
Abstract: We discuss a method of constructing solution of the initial value problem for duffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.
http://arxiv.org/abs/1102.4630

Title: The Riccati Differential Equation and a Diffusion-Type Equation
Authors: Erwin Suazo, Sergei K. Suslov, and Jose M. Vega-Guzman
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.
Published in New York Journal of Mathematics, Vol. 17a (2011):  http://nyjm.albany.edu:8000/j/2011/Vol17a.htm
http://nyjm.albany.edu:8000/j/2011/17a-14.html
http://arxiv.org/abs/0807.4349

Title: A Graphical Approach to a Model of Neuronal Tree with Variable Diameter
Authors: Marco Herrera-Valdez and Sergei K. Suslov
Abstract: We propose a simple graphical approach to steady state solutions of the cable equation for a general model of dendritic tree with tapering. A simple case of transient solutions is also briefly discussed.
http://arxiv.org/abs/1101.0296

Title: On Integrability of Nonautonomous Nonlinear Schroedinger Equations
Author: Sergei K. Suslov
Abstract:
We show, in general, how to transform nonautonomous nonlinear Schroedinger equation with quadratic Hamiltonians into the standard autonomous form that is completely integrable by the familiar inverse scattering method in nonlinear science. Derivation of the corresponding equivalent nonisospectral Lax pair is outlined.
http://arxiv.org/abs/1012.3661

Title: Soliton-like Solutions for Nonlinear Schroedinger Equation with Variable Quadratic Hamiltonians
Authors: Erwin Suazo and Sergei K. Suslov
Abstract:  We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schroedinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painleve transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons, and Jacobi elliptic and second Painleve transcendental solutions for several variable Hamiltonians that are important for current research in nonlinear optics and Bose-Einstein condensation. The Feshbach resonance matter wave soliton management is briefly discussed from this new perspective.
http://arxiv.org/abs/1010.2504

Title: Logistic Models with Time-Dependent Coefficients and Some of Their Applications
Authors:  Raquel M. Lopez, Benjamin R. Morin, Sergei K. Suslov
Abstract:  We discuss explicit solutions of the logistic model with variable parameters. Classical data on the sunflower seeds growth are revisited as a simple application of the logistic model with periodic coefficients. Some applications to related biological systems are briefly reviewed.
http://arxiv4.library.cornell.edu/abs/1008.2534v1

Title
: The Degenerate Parametric Oscillator and Ince's Equation
Comments: 9 pages, no figures, published in J. Phys. A: Math. Theor. http://iopscience.iop.org/1751-8121/44/1/015101
http://arxiv.org/abs/1006.3362
Abstract: We construct Green's function for the quantum degenerate parametric oscillator in terms of standard solutions of Ince's equation in a framework of a general approach to harmonic oscillators. Exact time-dependent wave functions and their connections with dynamical invariants and SU(1,1) group are also discussed.

Title: Dynamical Invariants for Variable Quadratic Hamiltonians
Author: Sergei K. Suslov
Comments: 21 pages, no figures, published in Physica Scripta:
http://iopscience.iop.org/1402-4896/81/5/055006
Abstract: We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamic invariants and solutions of the corresponding Cauchy initial value problem for the time-dependent Schroedinger equation are emphasized. An eigenfunction expansion of the solution of the initial value problem is also found. A nonlinear superposition principle for generalized Ermakov systems is established as a result of decomposition of the general quadratic invariant in terms of the linear ones. http://arxiv.org/abs/1002.0144

Title: Quantum Integrals of Motion for Variable Quadratic Hamiltonians
Authors: Ricardo Cordero-Soto, Erwin Suazo, and Sergei K. Suslov
Comments: 32 pages, no figures: Published in Annals of Physics
Abstract: We construct the integrals of motion for several models of the quantum damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the time-dependent Schroedinger equation with variable quadratic Hamiltonians. An extension of the Lewis-Riesenfeld dynamical invariant is given. The time-evolution of the expectation values of the energy related positive operators is determined for the oscillators under consideration. A proof of uniqueness of the corresponding Cauchy initial value problem is discussed as an application. http://arxiv.org/abs/0912.4900

Title: Mathematical Structure of Relativistic Coulomb Integrals
Author:
S. K. Suslov
published in Phys. Rev. A. 81 (2010), 032110; http://pra.aps.org/abstract/PRA/v81/i3/e032110)
Abstract:
We show that the diagonal 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, may be considered as difference analogs of the radial wave functions. Such structure provides an independent way of obtaining closed forms of these matrix elements by elementary methods of the theory of difference equations without explicit evaluation of the integrals. Three-term recurrence relations for each of these expectation values are derived as a by-product. Transformation formulas for the corresponding generalized hypergeometric series are discussed. http://arxiv.org/abs/0911.0111

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

Title: Relativistic Kramers-Pasternack Recurrence Relations
Author: Sergei K. Suslov
Comment: published in J. Phys. B: At Mol. Opt. Phys. 43 (2010) 074006 (7pp),  Special Issue on High Precision Atomic Physics.
http://iopscience.iop.org/0953-4075/43/7/074006
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.
http://arxiv.org/abs/0908.3021

Title: 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.

Title: Expectation Values $<r^p>$ for Harmonic Oscillator in $R^n$
Authors: Ricardo Cordero-Soto and Sergei K. Suslov
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.
http://arxiv.org/abs/0908.0032

Title: Expectation Values in Relativistic Coulomb Problems
Author: Sergei K. Suslov
Comments: published in J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 185003 (8pp) doi: 10.1088/0953-4075/42/18/185003
http://stacks.iop.org/0953-4075/42/185003
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.
http://arxiv.org/abs/0906.3338

Title: Models of Damped Oscillators in Quantum Mechanics
Authors: Ricardo Cordero-Soto, Erwin Suazo, and Sergei K. Suslov
Comments: Published in Journal of Physical Mathematics, Vol.1 (2009), Article ID S090603,16 pages, doi:10.4303/jpm/S090603
http://www.ashdin.com/journals/jpm/2009/S090603.htm
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.
http://arxiv.org/abs/0905.0507

Title: The time-dependent Schroedinger equation, Riccati equation and Airy functions
Authors: Nathan Lanfearand  Sergei K. Suslov
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$in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schroedinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is stablished. http://arxiv.org/abs/0903.3608 Title: Time Reversal for Modified Oscillators Authors: Ricardo Cordero-Soto and Sergei K. Suslov Comments: 33 pages, four figures, and two tables; published in Teoretical and Mathematical Physics 162 (2010)# 3, 345-380: http://83.149.209.141/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=6475&option_lang=eng Abstract: We discuss a new completely integrable case of the time-dependent Schroedinger equation in$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.
http://arxiv.org/abs/0808.3149

Title: An Integral Form of the Nonlinear Schroedinger Equation with Variable Coefficients
Authors: Erwin Suazo and Sergei Suslov
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.
http://arxiv.org/abs/0805.0633

Title: Solution of the Cauchy problem for a time-dependent Schroedinger equation
Authors: Maria Meiler, Ricardo Cordero-Soto, and Sergei K. Suslov
Comments: J. Math. Phys. 49 (2008)\#7, 072102 (27pp); published on line 9 July 2008:
Abstract: We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schroedinger 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.

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

Title: The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems
Authors: Sergei K. Suslov and Benjamin Trey
Comments: published in J. Math. Phys. 49,(2008) 012104 (51pp) ; DOI:10.1063/1.2830804
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.

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.
Comments: I has been one of the coordinating editors of an important 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, Revista Mexicana de Fisica, vol. 55, pp. 195--215, December 2009; http://rmf.fciencias.unam.mx/search/results?author=Suslov&=Search&search=1
http://arxiv.org/abs/0707.1902

Other Selected Publications:

1.   A.F. Nikiforov, S.K. Suslov, and V.B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow, 1985; English translation in Springer Series in Computational Physics, Springer-Verlag, 1991, MR1149380.

2.   R. Askey, M. Rahman, and S.K. Suslov, On a general q-Fourier transformation with nonsymmetric kernels, Journal of Computational and Applied Mathematics 68 (1996), 25-55, MR1418749.

3.  S.K. Suslov, An Introduction to Basic Fourier Series, Springer/Kluwer Series "Developments in Mathematics", Vol. 9, Springer/Kluwer Academic Publishers, Dordrecht - Boston- London, 2003, MR1978912.

4.  R. Cordero-Soto, R.M. Lopez, E. Suazo and S.K. Suslov, Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields, Lett. Math. Phys. 84 (2008)#2-3, 159-178, MR2415547.

5. S.K. Suslov, Mathematical structure of relativistic Coulomb integrals, Phys. Rev. A. 81 (2010), 032110; http://pra.aps.org/abstract/PRA/v81/i3/e032110