59 research outputs found

    On the spectra of infinite-dimensional Jacobi matrices

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    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

    Rotation Number Associated with Difference Equations Satisfied by Polynomials Orthogonal on the Unit Circle

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    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

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    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

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    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

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    We consider unitary analogs of dd-dimensional Anderson models on l2(Zd)l^2(\Z^d) defined by the product Uω=DωSU_\omega=D_\omega S where SS is a deterministic unitary and DωD_\omega is a diagonal matrix of i.i.d. random phases. The operator SS 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 Uω(Uωz)1U_\omega(U_\omega -z)^{-1}, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of SS. Such estimates imply almost sure localization for UωU_\omega

    Trace Formulas in Connection with Scattering Theory for Quasi-Periodic Background

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    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

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    We consider unitary analogs of 11-dimensional Anderson models on l2(Z)l^2(\Z) defined by the product Uω=DωSU_\omega=D_\omega S where SS is a deterministic unitary and DωD_\omega is a diagonal matrix of i.i.d. random phases. The operator SS 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 UωU_\omega is pure point almost surely for all values of the parameter of SS. We provide similar results for unitary operators defined on l2(N)l^2(\N) 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

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    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

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    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 (t=0t=0) and the triangular lattice (t=1t=1), and we obtain the asymptotics for 0<t10<t\le 1. For 0<t<10<t<1 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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