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

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

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

We show that the diagonal matrix elements $,$ 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.Comment: 13 pages, no figure

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

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

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

We investigate the spectral theory of the invariant Landau Hamiltonian
\La^\nu acting on the space ${\mathcal{F}}^\nu_{\Gamma,\chi}$ of
$(\Gamma,\chi)$-automotphic functions on \C^n, for given real number $\nu>0$,
lattice $\Gamma$ of \C^n and a map $\chi:\Gamma\to U(1)$ such that the
triplet $(\nu,\Gamma,\chi)$ 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 $\lambda=l=0,1,2, ...$. In such
case, ${\mathcal{E}}^\nu_{\Gamma,\chi}(l)$ is a finite dimensional vector space
whose the dimension is given explicitly. We show also that the eigenspace
${\mathcal{E}}^\nu_{\Gamma,\chi}(0)$ 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
$\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} +
\nu \bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$
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

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

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

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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