Second-Order Differential Operators in the Limit Circle Case

TL;DR

This paper develops a detailed theory for symmetric second-order differential operators in the limit circle case, introducing a quasiresolvent to explicitly describe all self-adjoint extensions and their spectral measures.

Contribution

It introduces the quasiresolvent concept and derives explicit formulas for resolvents and spectral measures, extending Jacobi operator theory to differential operators.

Findings

Explicit resolvent formulas for all self-adjoint realizations

Representation of spectral measures via Cauchy-Stieltjes transforms

Analogy with Jacobi operator spectral theory

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.