INITIAL VALUE PROBLEMS AND WEYL-TITCHMARSH THEORY FOR SCHRODINGER OPERATORS WITH OPERATOR-VALUED POTENTIALS

    Authors

    F. Gesztesy; R. Weikard;M. Zinchenko

    Comments

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    Abbreviated Journal Title

    Oper. Matrices

    Keywords

    Weyl-Titchmarsh theory; ODEs with operator coefficients; Schrodinger; operators; ABSOLUTELY CONTINUOUS-SPECTRUM; STURM-LIOUVILLE EQUATION; BOUNDARY-VALUE; PROBLEM; 2ND-ORDER ELLIPTIC OPERATOR; ORDER DIFFERENTIAL OPERATOR; SELF-ADJOINT EXTENSIONS; INFINITE BOUNDARIES; UNITARY EQUIVALENCE; ANALYTIC-FUNCTIONS; HERGLOTZ FUNCTIONS; Mathematics

    Abstract

    We develop Weyl-Titchmarsh theory for self-adjoint Schrodinger operators Ha in L-2((a,b); dx; H) associated with the operator-valued differential expression tau = -(d(2)/dX(2)) + V(.), with V: (a, b) - > B(H), and H a complex, separable Hilbert space. We assume regularity of the left endpoint a and the limit point case at the right endpoint b. In addition, the bounded self-adjoint operator a = a* is an element of B(H) is used to parametrize the self-adjoint boundary condition at the left endpoint a of the type sin(alpha)u' (a)+cos(a)u(a)= 0, with u lying in the domain of the underlying maximal operator H-max in L-2((a,b);dx; H) associated with t. More precisely, we establish the existence of the Weyl-Titchmarsh solution of H-alpha, the corresponding Weyl-Titchmarsh m-function m(alpha) and its Herglotz property, and determine the structure of the Green's function of H-alpha. Developing Weyl-Titchmarsh theory requires control over certain (operator-valued) solutions of appropriate initial value problems. Thus, we consider existence and uniqueness of solutions of 2nd-order differential equations with the operator coefficient V, {-y '' + (V-z)y = f on (a,b), y(x(0)) = h(0), y' (x(0)) = h(1), under the following general assumptions: (a, b) C R is a finite or infinite interval, x(0) is an element of (a, b), z is an element of C, V: (a,b) - > B(H) is a weakly measurable operator-valued function with parallel to V(.)parallel to(B(H)) is an element of L-loc(1) ((a,b);dx), and f is an element of L-loc(1) ((a,b);dx;H). We also study the analog of this initial value problem with y and f replaced by operator-valued functions Y, F is an element of B(H). Our hypotheses on the local behavior of V appear to be the most general ones to date.

    Journal Title

    Operators and Matrices

    Volume

    7

    Issue/Number

    2

    Publication Date

    1-1-2013

    Document Type

    Article

    Language

    English

    First Page

    241

    Last Page

    283

    WOS Identifier

    WOS:000322355900001

    ISSN

    1846-3886

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