Boundary triples and Weyl m-functions for powers of the Jacobi differential operator

[摘要] The abstract theory of boundary triples is applied to the classical Jacobi differential operator and its powers in order to obtain the Weyl m-function for several self-adjoint extensions with interesting bound- ary conditions: separated, periodic and those that yield the Friedrichs extension. These matrix-valued Nevanlinna-Herglotz m-functions are, to the best knowledge of the author, the first explicit examples to stem from singular higher-order differential equations. The creation of the boundary triples involves taking pieces, determined in [26], of the principal and non-principal solutions of the differential equation and putting them into the sesquilinear form to yield maps from the maximal domain to the boundary space. These maps act like quasi-derivatives, which are usually not well-defined for all functions in the maximal domain of singular expressions. However, well- defined regularizations of quasi-derivatives are produced by putting the pieces of the non-principal solutions through a modified Gram-Schmidt process. (c) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-11-05

[关键词] Boundary triples;Self-adjoint extension theory;Singular Sturm-Liouville operators;Nevanlinna-Herglotz;functions;Weyl m-functions