We would like to invite you to the second seminar of Quantum Circle ONLINE. Through the Zoom platform will speak Dale Frymark on Singular boundary conditions for Sturm-Liouville operators via perturbation theory. The seminar takes place on Tuesday, November 24, 2020 from 2.45 pm. The Zoom link, which will allow you to participate in the seminar, can be found on the Quantum Circle website.
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
Aθ = A0 + BθB*;
where A0 is a distinguished self-adjoint extension and Theta is a self-adjoint linear relation in Cd. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to A0, i.e. it belongs to H_1(A0). 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 θ.
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.