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

Abstract

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}_Θ=\boldsymbol{A}_0+{\bf B}Θ{\bf B}^*, \end{align*} where ${\mathbf{A}}_0$ 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 ${\mathbf{A}}_0$, i.e. it belongs to ${\mathcal{H}}_{-1}({\mathbf{A}}_0)$. 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.

29 pages, 1 figure

References in corpus (1)

• Weyl-Titchmarsh Theory for Sturm-Liouville Operators with Distributional Potentials

Cited by in corpus (3)

• Spectral Properties of Singular Sturm-Liouville Operators via Boundary Triples and Perturbation Theory
• The Krein-von Neumann extension revisited
• Perspectives on General Left-Definite Theory