Second-Order Differential Operators in the Limit Circle Case
Classical Analysis and ODEs
2021-08-17 v2
Abstract
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.
Cite
@article{arxiv.2105.08641,
title = {Second-Order Differential Operators in the Limit Circle Case},
author = {Dmitri R. Yafaev},
journal= {arXiv preprint arXiv:2105.08641},
year = {2021}
}