Boundary Value Problems, Weyl Functions, and Differential Operators

By Jussi Behrndt, S. Hassi, H. D. de Snoo. Monographs in Mathematics, 2020

189 citations recorded by the source corpus.

The book under review is a comprehensive treatment of boundary value problems from the unifying point of view of boundary triplets, offering a panoramic view ranging from abstract problems to concrete examples of ordinary and partial differential operators. Briefly and from a purely abstract point of view, the notion of boundary triplets, in the simplest and rather special context of self-adjoint problems, centers around a densely defined, closed, symmetric operator S in a complex Hilbert space H (often called the minimal operator in concrete applications) with equal and nonzero deficiency indices and its self-adjoint extensions denoted by SΘ, with Θ an appropriate operator (resp., relation) parameter indexing the self-adjoint extensions of S (see (3)). Thus, self-adjoint extensions of S can equivalently be viewed as self-adjoint restrictions of the adjoint S∗ of S; S∗ is typically called the maximal operator in applications. Given an auxiliary complex Hilbert space G, one seeks surjective boundary maps Γj : dom(S) → G, j = 0, 1, such that an abstract second Green identity holds in the form

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dede178868ee940ce6b3d4f093be8f1c96797986