We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more 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 the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalizat... more Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity-the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system-the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) A [cr(z)p(z)~l+2p(z)y(z)=O ' (1) Vxl(z) (2) A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89 and AT(z) = Vf(z + 1) = f(z + 1)-f(z).
We derive the recurrence relations for relativistic Coulomb integrals directly from the integral ... more We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.
We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödi... more We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n wit... more We discuss a new completely integrable case of the time-dependent Schrödinger 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 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 nonlinear Schrödinger equations are obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
We introduce a double sum extension of a very well-poised series and extend to this the transform... more We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6 û 5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.
We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic ... more 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.
We transform the time-dependent Schrödinger equation for the most general variable quadratic Hami... more We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.
Certain q-Fourier expansions found in the previous chapter give us a possibility to introduce ana... more Certain q-Fourier expansions found in the previous chapter give us a possibility to introduce analogs of the Bernoulli polynomials and numbers, the Euler polynomials and numbers, and the Riemann zeta function [146]. We shall study some of their properties that are close to the classical ones.
We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the... more We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system. Recent reports on observations of the dynamical Casimir effect [27], [51] strengthens the interest to 'nonclassical' states in quantum optics and generalized harmonic oscillators [10], [11], [13], [14], [15], [17], [33], [34] and [37]. The amplification of quantum fluctuations by modulating parameters of an oscillator is closely related to the process of particle production in quantum fields [11], [24], [34] and [37]. Other dynamical amplification mechanisms include the Unruh effect [47] and Hawking radiation [20], [21]. The purpose of this paper is to evaluate the Berry phase for certain "missing" solutions of the time-dependent Schrödinger equation for the linear harmonic oscillator as an instructive example. Applications will be discussed elsewhere.
With the help of computer algebra we study the diagonal matrix elements Or p , where O = {1, β, i... more With the help of computer algebra we study the diagonal matrix elements Or p , where O = {1, β, iαnβ} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem. Using Zeilberger's extension of Gosper's algorithm and a variant to it, three-term recurrence relations for each of these expectation values are derived together with some transformation formulas for the corresponding generalized hypergeometric series. In addition, the virial recurrence relations for these integrals are also found and proved algorithmically.
In this chapter we shall present elementary results on the convergence of the basic Fourier serie... more In this chapter we shall present elementary results on the convergence of the basic Fourier series, including a proof of completeness of the basic trigonometric systems, and will establish several convenient tools, such as asymptotics of zeros, which are important for practical investigation of these series in the next chapters. Methods of summation and a few explicit examples of q-Fourier series will be also discussed among other things.
Our main objective in this chapter is to derive several important integrals and expansion formula... more Our main objective in this chapter is to derive several important integrals and expansion formulas involving basic trigonometric functions on a q-quadratic grid. These results were established by Ismail, Rahman, Stanton, and Zhang in [68], [69], [71], and [76].
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more 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 the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalizat... more Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity-the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system-the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) A [cr(z)p(z)~l+2p(z)y(z)=O ' (1) Vxl(z) (2) A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89 and AT(z) = Vf(z + 1) = f(z + 1)-f(z).
We derive the recurrence relations for relativistic Coulomb integrals directly from the integral ... more We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.
We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödi... more We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n wit... more We discuss a new completely integrable case of the time-dependent Schrödinger 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 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 nonlinear Schrödinger equations are obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
We introduce a double sum extension of a very well-poised series and extend to this the transform... more We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6 û 5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.
We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic ... more 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.
We transform the time-dependent Schrödinger equation for the most general variable quadratic Hami... more We transform the time-dependent Schrödinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.
Certain q-Fourier expansions found in the previous chapter give us a possibility to introduce ana... more Certain q-Fourier expansions found in the previous chapter give us a possibility to introduce analogs of the Bernoulli polynomials and numbers, the Euler polynomials and numbers, and the Riemann zeta function [146]. We shall study some of their properties that are close to the classical ones.
We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the... more We evaluate the Berry phase for a "missing" family of the square integrable wavefunctions for the linear harmonic oscillator, which cannot be derived by the separation of variables (in a natural way). Instead, it is obtained by the action of the maximal kinematical invariance group on the standard solutions. A simple closed formula for the phase (in terms of elementary functions) is found by integration with the help of a computer algebra system. Recent reports on observations of the dynamical Casimir effect [27], [51] strengthens the interest to 'nonclassical' states in quantum optics and generalized harmonic oscillators [10], [11], [13], [14], [15], [17], [33], [34] and [37]. The amplification of quantum fluctuations by modulating parameters of an oscillator is closely related to the process of particle production in quantum fields [11], [24], [34] and [37]. Other dynamical amplification mechanisms include the Unruh effect [47] and Hawking radiation [20], [21]. The purpose of this paper is to evaluate the Berry phase for certain "missing" solutions of the time-dependent Schrödinger equation for the linear harmonic oscillator as an instructive example. Applications will be discussed elsewhere.
With the help of computer algebra we study the diagonal matrix elements Or p , where O = {1, β, i... more With the help of computer algebra we study the diagonal matrix elements Or p , where O = {1, β, iαnβ} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem. Using Zeilberger's extension of Gosper's algorithm and a variant to it, three-term recurrence relations for each of these expectation values are derived together with some transformation formulas for the corresponding generalized hypergeometric series. In addition, the virial recurrence relations for these integrals are also found and proved algorithmically.
In this chapter we shall present elementary results on the convergence of the basic Fourier serie... more In this chapter we shall present elementary results on the convergence of the basic Fourier series, including a proof of completeness of the basic trigonometric systems, and will establish several convenient tools, such as asymptotics of zeros, which are important for practical investigation of these series in the next chapters. Methods of summation and a few explicit examples of q-Fourier series will be also discussed among other things.
Our main objective in this chapter is to derive several important integrals and expansion formula... more Our main objective in this chapter is to derive several important integrals and expansion formulas involving basic trigonometric functions on a q-quadratic grid. These results were established by Ismail, Rahman, Stanton, and Zhang in [68], [69], [71], and [76].
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Papers by Sergei Suslov