The asymptotical spectrum of Jacobi matrices

摘要

A method to calculate the asymptotical eigenvalue density (asymptotical density of zeros) ρ(x) of Jacobi matrices (orthogonal polynomials) in terms of its moments is presented. This method does not require the convergence of continued fractions and inversion of functional transformations as previous ones do. It is shown to be applicable to a wide family of Jacobi matrices (orthogonal polynomials). As a byproduct the density ρ(x) is explicitly found for certain classical orthogonal polynomials.