1,920 research outputs found
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
Right limits and reflectionless measures for CMV matrices
We study CMV matrices by focusing on their right-limit sets. We prove a CMV
version of a recent result of Remling dealing with the implications of the
existence of absolutely continuous spectrum, and we study some of its
consequences. We further demonstrate the usefulness of right limits in the
study of weak asymptotic convergence of spectral measures and ratio asymptotics
for orthogonal polynomials by extending and refining earlier results of
Khrushchev. To demonstrate the analogy with the Jacobi case, we recover
corresponding previous results of Simon using the same approach
Multiple Meixner-Pollaczek polynomials and the six-vertex model
We study multiple orthogonal polynomials of Meixner-Pollaczek type with
respect to a symmetric system of two orthogonality measures. Our main result is
that the limiting distribution of the zeros of these polynomials is one
component of the solution to a constrained vector equilibrium problem. We also
provide a Rodrigues formula and closed expressions for the recurrence
coefficients. The proof of the main result follows from a connection with the
eigenvalues of block Toeplitz matrices, for which we provide some general
results of independent interest.
The motivation for this paper is the study of a model in statistical
mechanics, the so-called six-vertex model with domain wall boundary conditions,
in a particular regime known as the free fermion line. We show how the multiple
Meixner-Pollaczek polynomials arise in an inhomogeneous version of this model.Comment: 32 pages, 4 figures. References adde
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