Sergei K. Suslov

On classical orthogonal polynomials

Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.

Propagator of a charged particle with a spin in uniform magnetic and perpendicular electric fields

We construct an explicit solution of the Cauchy initial value problem for the time-dependent Schrödinger 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 Schrödinger equation and some special and limiting cases are outlined.

An introduction to basic Fourier series

Foreword. Preface. 1: Introduction. 2: Basic Exponential and Trigonometric Functions. 3: Addition Theorems. 4: Some Expansions and Integrals. 5: Introduction of Basic Fourier Series. 6: Investigation of Basic Fourier Series. 7: Completeness of Basic Trigonometric Systems. 8: Improved Asymptotics of Zeros. 9: Some Expansions in Basic Fourier Series. 10: Basic Bernoulli and Euler Polynomials and Numbers and q-Zeta Function. 11: Numerical Investigation of Basic Fourier Series. 12: Suggestions for Further Work. Appendix A: Selected Summation and Transformation Formulas and Integrals. A.1. Basic Hypergeometric Series. A.2. Selected Summation Formulas. A.3. Selected Transformation Formulas. A.4. Some Basic Integrals. Appendix B: Some Theorems of Complex Analysis. B.1. Entire Functions. B.2. Lagrange Inversion Formula. B.3. Dirichlet Series. B.4. Asymptotics. Appendix C: Tables of Zeros of Basic Sine and Cosine Functions. Appendix D: Numerical Examples of Improved Asymptotics. Appendix E: Numerical Examples of Euler-Rayleigh Method. Appendix F: Numerical Examples of Lower and Upper Bounds. Bibliography. Index.

Special functions 2000 : current perspective and future directions

Preface. Foreword. Baileys transform, lemma, chains and tree G.E. Andrews. Riemann-Hilbert problems for multiple orthogonal polynomials W. Van Assche, et al. Flowers which we cannot yet see growing in Ramanujans garden of hypergeometric series, elliptic functions and qs B.C. Berndt. Orthogonal rational functions and continued fractions A. Bultheel, et al. Orthogonal polynomials and reflection groups C.F. Dunkl. The bispectral problem: an overview F.A. Grunbaum. The Bochner-Krall problem: some new perspectives L. Haine. Lectures on q-orthogonal polynomials M.E.H. Ismail. The Askey-Wilson function transform scheme E. Koelink, J.V. Stokman. Arithmetic of the partition function K. Ono. The associated classical orthogonal polynomials M. Rahman. Special functions defined by analytic difference equations S.N.M. Ruijsenaars. The factorization method, self-similar potentials and quantum algebras V.P. Spiridonov. Generalized eigenvalue problem and a new family of rational functions biorthogonal on elliptic grids V.P. Spiridonov, A.S. Zhedanov. Orthogonal polynomials and combinatorics D. Stanton. Basic exponential functions on a q-quadratic grid S.K. Suslov. Projection operator method for quantum groups V.N. Tolstoy. Uniform asymptotic expansions R. Wong. Exponential asymptotics R. Wong. Index.

Quantum integrals of motion for variable quadratic Hamiltonians

Abstract We construct integrals of motion for several models of the quantum damped oscillators in a framework of a general approach to the time-dependent Schrodinger 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.

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

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrodinger 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.

Dynamical invariants for variable quadratic Hamiltonians

We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value problem for the time-dependent Schrodinger 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.

Classical orthogonal polynomials of a discrete variable on nonuniform lattices

A general theory of classical orthogonal polynomials of a discrete variable on nonuniform lattices is developed. The classification of the polynomials under consideration is given.

Some Expansions in Basic Fourier Series and Related Topics

We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product.

Difference Hypergeometric Functions

The particular solutions for hypergeometric-type difference equations on non-uniform lattices are constructed by using the method of undetermined coefficients. Recently there has been revived interest in the classical theory of special functions. In particular, this theory has been further developed through studying their difference analogs [AWl], [AW2], [NU1], [NSU], [AS1], [AS2], and [S]. Quantum algebras [D], [VS], [KR], [K], which are being developed nowadays, provide a natural basis for the group-theoretic interpretation of these difference special functions. In the present paper we discuss a method of constructing the solutions for difference equations of hypergeometric type on non-uniform lattices [AS2].