47 research outputs found
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