Perspectives on General Left-Definite Theory

TL;DR

This paper reviews the development and connections of general left-definite theory with other self-adjoint extension theories, highlighting its ongoing relevance and potential for future research in differential operators.

Contribution

It compares and contrasts left-definite theory with BKV theory and singular perturbation theory, encouraging further exploration of their interrelations.

Findings

Left-definite theory has broad applications in differential operators.

Connections between left-definite, BKV, and singular perturbation theories are elucidated.

Open questions are proposed to guide future research.

Abstract

In 2002, Littlejohn and Wellman developed a celebrated general left-definite theory for semi-bounded self-adjoint operators with many applications to differential operators. The theory starts with a semi-bounded self-adjoint operator and constructs a continuum of related Hilbert spaces and self-adjoint operators that are intimately related with powers of the initial operator. The development spurred a flurry of activity in the field that is still ongoing today. The main goal of this expository (with the exception of Proposition 1) manuscript is to compare and contrast the complementary theories of general left-definite theory, the Birman--Krein--Vishik (BKV) theory of self-adjoint extensions and singular perturbation theory. In this way, we hope to encourage interest in left-definite theory as well as point out directions of potential growth where the fields are interconnected. We include several related open questions to further these goals.