Lowering and raising operators for the free Meixner class of orthogonal polynomials

We compare some properties of the lowering and raising operators for the classical and free classes of Meixner polynomials on the real line

On a two variable class of Bernstein-Szego measures

The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.Comment: minor change

Spectral and Localization Properties for the One-Dimensional Bernoulli Discrete Dirac Operator

A 1D Dirac tight-binding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schr?odinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases.Comment: no figure; 24 pages; to appear in Journal of Mathematical Physic

Intertwining technique for a system of difference Schroedinger equations and new exactly solvable multichannel potentials

The intertwining operator technique is applied to difference Schroedinger equations with operator-valued coefficients. It is shown that these equations appear naturally when a discrete basis is used for solving a multichannel Schroedinger equation. New families of exactly solvable multichannel Hamiltonians are found

The class of n-entire operators

We introduce a classification of simple, regular, closed symmetric operators with deficiency indices (1,1) according to a geometric criterion that extends the classical notions of entire operators and entire operators in the generalized sense due to M. G. Krein. We show that these classes of operators have several distinctive properties, some of them related to the spectra of their canonical selfadjoint extensions. In particular, we provide necessary and sufficient conditions on the spectra of two canonical selfadjoint extensions of an operator for it to belong to one of our classes. Our discussion is based on some recent results in the theory of de Branges spaces.Comment: 33 pages. Typos corrected. Changes in the wording of Section 2. References added. Examples added. arXiv admin note: text overlap with arXiv:1104.476

Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra U_q(su(1,1)) is studied. Spectrum and eigenfunctions of this operator are found in an explicit form. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial U_q(su(1,1))-basis and the basis of eigenfunctions are interrelated by a matrix with entries, expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials M_n(x;b,c;q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.Comment: 15 pages, LaTe

Spectral averaging techniques for Jacobi matrices with matrix entries

A Jacobi matrix with matrix entries is a self-adjoint block tridiagonal matrix with invertible blocks on the off-diagonals. Averaging over boundary conditions leads to explicit formulas for the averaged spectral measure which can potentially be useful for spectral analysis. Furthermore another variant of spectral averaging over coupling constants for these operators is presented

Boundary relations and generalized resolvents of symmetric operators

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in \dC_+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page

Discrete quantum model of the harmonic oscillator

We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bounded, whereas the spectra of position and momentum are a denumerable non-degenerate set of points in [-1,1] that depends on the deformation parameter q from (0,1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q-Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit q to 1, one recovers the ordinary quantum harmonic oscillator.Comment: 21 page

Discrete series representations for sl(2|1), Meixner polynomials and oscillator models

We explore a model for the one-dimensional quantum oscillator based upon the Lie superalgebra sl(2|1). For this purpose, a class of discrete series representations of sl(2|1) is constructed, each representation characterized by a real number beta>0. In this model, the position and momentum operators of the oscillator are odd elements of sl(2|1) and their expressions involve an arbitrary parameter gamma. In each representation, the spectrum of the Hamiltonian is the same as that of the canonical oscillator. The spectrum of the momentum operator can be continuous or infinite discrete, depending on the value of gamma. We determine the position wavefunctions both in the continuous and discrete case, and discuss their properties. In the discrete case, these wavefunctions are given in terms of Meixner polynomials. From the embedding osp(1|2)\subset sl(2|1), it can be seen why the case gamma=1 corresponds to the paraboson oscillator. Consequently, taking the values (beta,gamma)=(1/2,1) in the sl(2|1) model yields the canonical oscillator.Comment: (some minor misprints were corrected in this version