58 research outputs found
On the spectra of infinite-dimensional Jacobi matrices
AbstractThe Green's function method used by Case and Kac is extended to include unbounded Jacobi matrices. As a first application an upper bound on the number of eigenvalues is calculated, using the method of Bargmann. Another bound is found using the Birman-Schwinger argument, which is valid for matrix orthogonal polynomials
Corrigendum to “On the spectra of infinite-dimensional Jacobi matrices” [J. Approx. Theory 53 (1988) 251–265]
Rotation Number Associated with Difference Equations Satisfied by Polynomials Orthogonal on the Unit Circle
AbstractThe techniques of topological dynamics and differential-dynamical systems are used to study polynomials orthogonal with respect to a measure supported on the unit circle. It is assumed that the reflection coefficients associated with these polynomials form a stationary stochastic ergodic process. In particular, the techniques mentioned above are used to prove a gap labelling result
Orthogonal polynomials with asymptotically periodic recurrence coefficients
AbstractGiven the coefficients in the three term recurrence relation satisfied by orthogonal polynomials, we investigate the properties of those classes of polynomials whose coefficients are asymptotically periodic. Assuming a rate of convergence of the coefficients to their asymptotic values, we construct the measure with respect to which the polynomials are orthogonal and discuss their asymptotic behavior
Fractal Functions and Wavelet Expansions Based on Several Scaling Functions
AbstractWe present a method for constructing translation and dilation invariant functions spaces using fractal functions defined by a certain class of iterated function systems. These spaces generalize the C0 function spaces constructed in [D. Hardin, B. Kessler, and P. R. Massopust, J. Approx. Theory71 (1992), 104-120] including, for instance, arbitrarily smooth function spaces. These new function spaces are generated by several scaling functions and their integer-translates. We give necessary and sufficient conditions for these function spaces to form a multiresolution analysis of L2R
Fractional Moment Estimates for Random Unitary Operators
We consider unitary analogs of dimensional Anderson models on
defined by the product where is a deterministic
unitary and is a diagonal matrix of i.i.d. random phases. The
operator is an absolutely continuous band matrix which depends on
parameters controlling the size of its off-diagonal elements. We adapt the
method of Aizenman-Molchanov to get exponential estimates on fractional moments
of the matrix elements of , provided the
distribution of phases is absolutely continuous and the parameters correspond
to small off-diagonal elements of . Such estimates imply almost sure
localization for
Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background
We investigate trace formulas for Jacobi operators which are trace class
perturbations of quasi-periodic finite-gap operators using Krein's spectral
shift theory. In particular we establish the conserved quantities for the
solutions of the Toda hierarchy in this class.Comment: 7 page
Localization for Random Unitary Operators
We consider unitary analogs of dimensional Anderson models on
defined by the product where is a deterministic
unitary and is a diagonal matrix of i.i.d. random phases. The
operator is an absolutely continuous band matrix which depends on a
parameter controlling the size of its off-diagonal elements. We prove that the
spectrum of is pure point almost surely for all values of the
parameter of . We provide similar results for unitary operators defined on
together with an application to orthogonal polynomials on the unit
circle. We get almost sure localization for polynomials characterized by
Verblunski coefficients of constant modulus and correlated random phases
Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports
In this paper we present a survey about analytic properties of polynomials
orthogonal with respect to a weighted Sobolev inner product such that the
vector of measures has an unbounded support. In particular, we are focused in
the study of the asymptotic behaviour of such polynomials as well as in the
distribution of their zeros. Some open problems as well as some new directions
for a future research are formulated.Comment: Changed content; 34 pages, 41 reference
Asymptotics of block Toeplitz determinants and the classical dimer model
We compute the asymptotics of a block Toeplitz determinant which arises in
the classical dimer model for the triangular lattice when considering the
monomer-monomer correlation function. The model depends on a parameter
interpolating between the square lattice () and the triangular lattice
(), and we obtain the asymptotics for . For we apply the
Szeg\"o Limit Theorem for block Toeplitz determinants. The main difficulty is
to evaluate the constant term in the asymptotics, which is generally given only
in a rather abstract form
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