Linear Operators and Operator Functions Associated with Spectral Boundary Value Problems

TL;DR

This paper develops a comprehensive operator-theoretic framework for spectral boundary value problems, introducing generalized boundary conditions, solvability results, and explicit resolvent formulas with applications to PDEs and physics.

Contribution

It introduces an abstract spectral boundary value problem framework with generalized boundary conditions, solvability criteria, and explicit resolvent representations based on Krein's formula.

Findings

Representation of solutions using Krein's resolvent formula

Explicit description of operator domains under boundary conditions

Application of theory to PDEs and mathematical physics

Abstract

The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is suggested and results on its solvability complemented by representations of weak and strong solutions are obtained. Existence of a closed linear operator defined by a given boundary condition and description of its domain are studied in detail. These questions are addressed on the basis of Krein's resolvent formula derived from the explicit representations of solutions also obtained here. Usual resolvent identities for two operators associated with two different boundary conditions are written in terms of the so called M-function. Abstract considerations are complemented by illustrative examples taken from the theory of partial differential operators. Other applications to boundary value problems of analysis and mathematical physics are outlined. (Initial version title "Spectral Boundary Value Problems and their Linear Operators", 2009)