The limit-point/limit-circle classification for ordinary differential equations with distributional coefficients

TL;DR

This paper extends the classical limit-point/limit-circle classification to differential equations with distributional coefficients, identifying conditions under which traditional criteria apply or fail.

Contribution

It introduces a framework for classifying differential equations with distributional coefficients, clarifying when classical Weyl criteria are valid or need modification.

Findings

Classical Weyl alternative applies under specific distributional conditions.

Identifies cases where the classical classification fails due to distributional coefficients.

Provides a comprehensive analysis of limit-point/limit-circle for matrix-valued distributional coefficients.

Abstract

We investigate the limit-point/limit-circle classification for the differential equation $Ju'+qu=\lambda wu$ where $J=\big(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}\big)$ and $q$ and $w$ are matrices whose entries are distributions of order zero with $q$ Hermitian and $w$ non-negative. We identify the situations when the classical alternative of Weyl works and when it fails.