48 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
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
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
Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle
19 pages, no figures.-- MSC2000 codes: 42C05, 47A56.MR#: MR1970413 (2004b:42058)Zbl#: Zbl 1047.42021Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of L_n(\tilde{\Omega}) L_n(\Omega) -1} and \Phi_n(z, \tilde{\Omega}) \Phi_n(z, \tilde{\Omega}) -1} where , , M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln(·) means the leading coefficient of the orthonormal matrix polynomials Φn(z; •).Finally, we deduce the asymptotic behavior of in the case when M=I.The work of the second author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-C03-01 and INTAS
Project INTAS93-0219 Ext.Publicad
Subcritical multiplicative chaos for regularized counting statistics from random matrix theory
For an N×N random unitary matrix U_N, we consider the random field defined by counting the number of eigenvalues of U_N in a mesoscopic arc of the unit circle, regularized at an N-dependent scale Ɛ_N>0. We prove that the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. In addition, we show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. The proofs are based on the asymptotic analysis of certain Toeplitz or Fredholm determinants using the Borodin-Okounkov formula or a Riemann-Hilbert problem for integrable operators. Our approach to the L¹-phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context
Lyapunov stability analysis of higher-order 2D systems
We prove a necessary and sufficient condition for the asymptotic stability of a 2D system described by a system of higher-order linear partial difference equations. We show that asymptotic stability is equivalent to the existence of a vector Lyapunov functional satisfying certain positivity conditions together with its divergence along the system trajectories. We use the behavioral framework and the calculus of quadratic difference forms based on four-variable polynomial algebra