Boundary Conditions associated with the General Left-Definite Theory for Differential Operators

In the early 2000's, Littlejohn and Wellman developed a general left-definite theory for certain self-adjoint operators by fully determining their domains and spectral properties. The description of these domains do not feature explicit boundary conditions. We present characterizations of these domains given by the left-definite theory for all operators which possess a complete system of orthogonal eigenfunctions, in terms of classical boundary conditions.Comment: 28 page

Perspectives on General Left-Definite Theory

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

Boundary conditions associated with left-definite theory and the spectral analysis of iterated rank-one perturbations.

This dissertation details the development of several analytic tools that are used to apply the techniques and concepts of perturbation theory to other areas of analysis. The main application is an efficient characterization of the boundary conditions associated with the general left-definite theory for differential operators. This theory originated with the groundbreaking work of Littlejohn and Wellman in 2002 which fully determined the `left-definite domains' and spectral properties of powers of self-adjoint Sturm--Liouville operators associated with classical orthogonal polynomials. We will study how the left-definite domains associated with these operators can be explicitly described by classical boundary conditions. Additional applications are made to infinite rank perturbations by successively introducing rank-one perturbations to a self-adjoint operator with absolutely continuous spectrum. The absolutely continuous part of the spectral measure of the constructed operator is controlled and estimated

Singular Boundary Conditions for Sturm--Liouville Operators via Perturbation Theory

We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in\{1,2\}$. This characterization generalizes the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as \begin{align*} \boldsymbol{A}_\Theta=\boldsymbol{A}_0+{\bf B}\Theta{\bf B}^*, \end{align*} where $\boldsymbol{A}_0$ is a distinguished self-adjoint extension and $\Theta$ is a self-adjoint linear relation in $\mathbb{C}^d$. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to $\boldsymbol{A}_0$, i.e. it belongs to $\mathcal{H}_{-1}(\boldsymbol{A}_0)$. The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $\Theta$. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.Comment: 31 pages, 1 figur