Boundary Data Maps and Krein's Resolvent Formula for Sturm-Liouville Operators on a Finite Interval


We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval (a,b), to more general boundary conditions, began in [8] and [17]. While these earlier studies of boundary data maps focused on the case of general separated boundary conditions at a and b, the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones). In the course of this paper we describe the connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann's basic parametrization in terms of unitary maps between deficiency subspaces.


Mathematics and Statistics

Keywords and Phrases

Boundary data maps; Krein-type resolvent formulas; Parametrizations of self-adjoint extensions; Perturbation determinants; Self-adjoint Sturm-Liouville operators on a finite interval; Spectral shift functions

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Article - Journal

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Publication Date

01 Mar 2014