2,287 research outputs found

    Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice

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    We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on non-homogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org

    Arbitrary l-state solutions of the rotating Morse potential by the asymptotic iteration method

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    For non-zero \ell values, we present an analytical solution of the radial Schr\"{o}dinger equation for the rotating Morse potential using the Pekeris approximation within the framework of the Asymptotic Iteration Method. The bound state energy eigenvalues and corresponding wave functions are obtained for a number of diatomic molecules and the results are compared with the findings of the super-symmetry, the hypervirial perturbation, the Nikiforov-Uvarov, the variational, the shifted 1/N and the modified shifted 1/N expansion methods.Comment: 15 pages with 1 eps figure. accepted for publication in Journal of Physics A: Mathematical and Genera

    Mathematical Structure of Relativistic Coulomb Integrals

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    We show that the diagonal matrix elements ,, where OO =1,β,iαnβ={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.Comment: 13 pages, no figure

    Orthogonal polynomials of discrete variable and Lie algebras of complex size matrices

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    We give a uniform interpretation of the classical continuous Chebyshev's and Hahn's orthogonal polynomials of discrete variable in terms of Feigin's Lie algebra gl(N), where N is any complex number. One can similarly interpret Chebyshev's and Hahn's q-polynomials and introduce orthogonal polynomials corresponding to Lie superlagebras. We also describe the real forms of gl(N), quasi-finite modules over gl(N), and conditions for unitarity of the quasi-finite modules. Analogs of tensors over gl(N) are also introduced.Comment: 25 pages, LaTe

    Raising and lowering operators, factorization and differential/difference operators of hypergeometric type

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    Starting from Rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variable on uniform lattice. In order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight. Using the Infeld-Hull factorization method, we generate from the raising and lowering operators the second order self-adjoint differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission

    Landau (\Gamma,\chi)-automorphic functions on \mathbb{C}^n of magnitude \nu

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    We investigate the spectral theory of the invariant Landau Hamiltonian \La^\nu acting on the space FΓ,χν{\mathcal{F}}^\nu_{\Gamma,\chi} of (Γ,χ)(\Gamma,\chi)-automotphic functions on \C^n, for given real number ν>0\nu>0, lattice Γ\Gamma of \C^n and a map χ:ΓU(1)\chi:\Gamma\to U(1) such that the triplet (ν,Γ,χ)(\nu,\Gamma,\chi) satisfies a Riemann-Dirac quantization type condition. More precisely, we show that the eigenspace {\mathcal{E}}^\nu_{\Gamma,\chi}(\lambda)=\set{f\in {\mathcal{F}}^\nu_{\Gamma,\chi}; \La^\nu f = \nu(2\lambda+n) f}; \lambda\in\C, is non trivial if and only if λ=l=0,1,2,...\lambda=l=0,1,2, .... In such case, EΓ,χν(l){\mathcal{E}}^\nu_{\Gamma,\chi}(l) is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace EΓ,χν(0){\mathcal{E}}^\nu_{\Gamma,\chi}(0) associated to the lowest Landau level of \La^\nu is isomorphic to the space, {\mathcal{O}}^\nu_{\Gamma,\chi}(\C^n), of holomorphic functions on \C^n satisfying g(z+\gamma) = \chi(\gamma) e^{\frac \nu 2 |\gamma|^2+\nu\scal{z,\gamma}}g(z), \eqno{(*)} that we can realize also as the null space of the differential operator j=1n(2zjzˉj+νzˉjzˉj)\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + \nu \bar z_j \frac{\partial}{\partial \bar z_j}) acting on C\mathcal C^\infty functions on \C^n satisfying ()(*).Comment: 20 pages. Minor corrections. Scheduled to appear in issue 8 (2008) of "Journal of Mathematical Physics

    Polynomial Solutions of Shcrodinger Equation with the Generalized Woods Saxon Potential

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    The bound state energy eigenvalues and the corresponding eigenfunctions of the generalized Woods Saxon potential are obtained in terms of the Jacobi polynomials. Nikiforov Uvarov method is used in the calculations. It is shown that the results are in a good agreement with the ones obtained before.Comment: 14 pages, 2 figures, submitted to Physical Review

    Discrete Darboux transformation for discrete polynomials of hypergeometric type

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    Darboux Transformation, well known in second order differential operator theory, is applied here to the difference equation satisfied by the discrete hypergeometric polynomials(Charlier, Meixner-Krawchuk, Hahn)

    A search on the Nikiforov-Uvarov formalism

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    An alternative treatment is proposed for the calculations carried out within the frame of Nikiforov-Uvarov method, which removes a drawback in the original theory and by pass some difficulties in solving the Schrodinger equation. The present procedure is illustrated with the example of orthogonal polynomials. The relativistic extension of the formalism is discussed.Comment: 10 page
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