On Molchanov's criterion for compactness of the resolvent for a non self-adjoint Sturm-Liouville operator

TL;DR

This paper investigates the applicability and limitations of Molchanov's criterion for the compactness of the resolvent in Sturm-Liouville operators, clarifying conditions under which the criterion is valid or insufficient.

Contribution

It analyzes the validity of Molchanov's condition for Sturm-Liouville operators with potentials in specific sectors, extending understanding of when the criterion applies.

Findings

Molchanov's condition is necessary but not sufficient for Sturm-Liouville operators.

The criterion remains valid for potentials in narrower sectors away from the negative half-plane.

The work clarifies the limitations of Molchanov's criterion in the context of Sturm-Liouville operators.

Abstract

The Molchanov's condition is a necessary condition for the compactness of the resolvent for a wide class of ordinary differential operators of arbitrary order, but for the Sturm-Liouville operator it is not sufficient, even if the real part of the potential is non-negative. Molchanov's criterion remains valid in the formulation closest to the original one for potentials that take values in a narrower sector than the half-plane, separated from the negative half-axis. This work is devoted to these questions.