Dimitri R. Yafaev

Spectral properties of Jacobi operators J are intimately related to an asymptotic behavior of the corresponding orthogonal polynomials P_{n}(z) as n\to\infty . We study the case where the off-diagonal coefficients a_{n} and, eventually, diagonal coefficients b_{n} of J tend to infinity in such a way that the ratio \gamma_{n}\coloneq 2^{-1}b_{n}(a_{n}a_{n-1})^{-1/2} has a finite limit \gamma . In the case |\gamma | < 1 asymptotic formulas for P_{n}(z) generalize those for the Hermite polynomials and the corresponding Jacobi operators J have absolutely continuous spectra covering the whole real line. If |\gamma | > 1 , then spectra of the operators J are discrete. Our goal is to investigate the critical case | \gamma |=1 that occurs, for example, for the Laguerre polynomials. The formulas obtained depend crucially on the rate of growth of the coefficients a_{n} (or b_{n} ) and are qualitatively different in the cases where a_{n}\to \infty faster or slower than n . For the fast growth of a_{n} , we also have to distinguish the cases |\gamma_{n}| \to 1-0 and |\gamma_{n}| \to 1+0 . Spectral properties of the corresponding Jacobi operators are quite different in all these cases. Our approach works for an arbitrary power growth of the Jacobi coefficients.

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.