Conditions for sectoriality and compactness of the resolvent for a non-self-adjoint Sturm--Liouville operator with singular distributional potential

TL;DR

The paper addresses spectral properties of non-self-adjoint Sturm–Liouville operators with complex distributional potentials by expressing q as the distributional derivative of an antiderivative s∈L_{2,loc}(R_+) and regularizing the operator via y^{[1]}=y'-sy. It develops a cohesive framework—via a necessary blow-up condition on the antiderivative, Ismagilov's localization principle, and multiple sectoriality/compact-resolvent criteria—to characterize when the minimal operator admits extensions with a compact resolvent. It further investigates the Miura-type representation of s, proves the semi-boundedness criterion, and constructs counterexamples showing limits of homogeneity and prior conditions under perturbations. Overall, the results extend Molchanov-type criteria to complex-valued potentials, yielding sharp, locality-based conditions for sectoriality and resolvent compactness in singular Sturm–Liouville problems.

Abstract

The aim of this paper is to find necessary and sufficient conditions for sectoriality and compactness of the resolvent for Sturm--Liouville operators with complex-valued potentials of the class $q\in W_{2,loc}^{-1}(\mathbb{R}_+)$ in terms of its generalized antiderivatives $s\in L_{2,loc}(\mathbb{R}_+)$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Theorem 6
• Lemma 2: Ismagilov's localization principle
• Lemma 3
• Lemma : Rellich