Comparative asymptotics of solutions and trace formulas for a class of difference equations. (English. Russian original) Zbl 1287.39010
Proc. Steklov Inst. Math. 272, Suppl. 2, S96-S137 (2011); translation from Sovrem. Probl. Mat. 6, 4-74 (2006).
Summary: 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 singular integral equation, studied by J. Nuttal [Constructive Approximation 6, No. 2, 157–166 (1990; Zbl 0685.41014)].
MSC:
| 39A22 | Growth, boundedness, comparison of solutions to difference equations |
| 39A10 | Additive difference equations |
Keywords:
Jacobi operator; Markov function; spectral measure; asymptotic formula; difference equation; Weyl solution; trace formula; singular integral equationCitations:
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