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

    Authors

    F. Gesztesy; R. Weikard;M. Zinchenko

    Comments

    Authors: contact us about adding a copy of your work at [email protected]

    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

    Share

    COinS
     

    Accessibility Statement

    This item was created or digitized prior to April 24, 2027, or is a reproduction of legacy media created before that date. It is preserved in its original, unmodified state specifically for research, reference, or historical recordkeeping. In accordance with the ADA Title II Final Rule, the University Libraries provides accessible versions of archival materials upon request. To request an accommodation for this item, please submit an accessibility request form.