Comparative Asymptotic Behavior of Solutions and Trace Formulas for a Class of Difference Equations

Properties of Jacobi operators generated by Markov functions are studied. The main results refer to the case where the support of the corresponding spectral measure $\mu$ consists of several intervals of the real line. In this class of operators, a comparative asymptotic formula for two solutions of the corresponding difference equation, polynomials orthogonal with respect to the measure $\mu$ and functions of the second kind (Weyl solutions) is found. Asymptotic trace formulas for the coefficients $a_n$ and $b_n$ in this difference equation are obtained. The derivation of the asymptotic formulas is based on standard techniques for studying the asymptotic properties of polynomials orthogonal on several intervals and consists in reducing the asymptotic problem to investigating properties of solutions to the Nuttall singular integral equation.