Matrix-valued spectral theory applied to various differential operators
Date
2022
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Publisher
University of Delaware
Abstract
The theory of finite-rank self-adjoint perturbations allows for the determination of spectral information for otherwise inaccessible classes of operators by using the tools of analytic function theory in the matrix-valued setting. The work presented in this dissertation explores several applications of this theory. More specifically, we obtain matrix-valued spectral measures for families of differential operators that fall into two categories: derivative powers and singular Sturm--Liouville operators. ☐ Via the connection between matrix-valued contractive analytic functions and matrix-valued nonnegative measures through the Herglotz Representation Theorem, we use matrix-valued Clark theory to determine the support and weights of matrix-valued spectral measures for derivative powers. For several powers of the derivative operator, explicit expressions are included. While eigenfunctions and eigenvalues for these operators with fixed boundary conditions can often be computed using direct methods from ordinary differential equations, this approach provides a more complete picture of the spectral information. Matrix-valued spectral measures come with a notion of directionality in spectral information, which is not present when using scalar-valued spectral measures. As examples, we compute information for rank one and two cases on both the half-line and finite intervals. ☐ Through marrying the theories of boundary triples and finite-rank perturbations, we obtain a more holistic view of the spectral information of self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) than what either theory describes in isolation. We show that these operators can be expressed as additive singular form bounded self-adjoint perturbations, generalizing the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. 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. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and analyzed with tools from both theories.
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Keywords
Matrix-valued spectral theory