Mark Krein's Method of Directing Functionals and Singular Potentials

TL;DR

This paper demonstrates how Krein's method of directing functionals can establish the existence of scalar spectral measures for specific Sturm-Liouville problems with singular endpoints, under certain solution conditions.

Contribution

It applies Krein's directing functionals method to prove spectral measure existence for Sturm-Liouville equations with singular endpoints, expanding its applicability.

Findings

Existence of scalar spectral measure established for certain Sturm-Liouville problems.

Utilizes solutions that are square integrable at one endpoint and depend analytically on eigenvalues.

Provides a new approach to spectral analysis of singular differential operators.

Abstract

It is shown that M. Krein's method of directing functionals can be used to prove the existence of a scalar spectral measure for certain Sturm-Liouville equations with two singular endpoints. The essential assumption is the existence of a solution of the equation that is square integrable at one singular endpoint and depends analytically on the eigenvalue parameter.