Dale Frymark: Characterisations and Decompositions of Domains for Powers of Classical Sturm-Liouville Operators

Tid: On 2019-02-13 kl 13.15 - 14.15

Föreläsare: Dale Frymark (SU)

Plats: Room 306, House 6, Kräftriket, Department of Mathematics, Stockholm University

Abstract: We set out to build a framework for self-adjoint extension theory when the operators are powers of classical Sturm-Liouville operators. This is done by analysing the structures of maximal and minimal domains, as well as the defect spaces. We focus to some extent on the Jacobi differential operator, with restrictions for the Laguerre operator.
The maximal domain for compositions of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the underlying Hilbert space can then be decomposed into powers of (1-x) and (1+x). Most of these powers are shown to be in the associated
minimal domain, leading to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are given in terms of the new basis functions for the defect spaces using Glazman-Krein-Naimark theory.