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=1 or 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 $$ A_{\Theta} = A_0 + B \Theta B*, $$ where $A_0$ 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 $A_0$, i.e. it belongs to $H_{-1}(A_0)$. 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 is analyzed with tools both from the theory of boundary triples and perturbation theory.

mathematical physicsclassical analysis and ODEs

Audience: researchers in the topic

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