Tuesday, April 17, 2012

1204.3314 (Stephen Clark et al.)

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

Stephen Clark, Fritz Gesztesy, Roger Nichols, Maxim Zinchenko
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. While 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.
View original: http://arxiv.org/abs/1204.3314

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