Boundary Triples and Weyl $m$-functions for Powers of the Jacobi Differential Operator

TL;DR

This paper applies boundary triple theory to the Jacobi differential operator and its powers, deriving explicit matrix-valued Weyl m-functions for various self-adjoint extensions, including singular higher-order cases.

Contribution

It provides the first explicit examples of Weyl m-functions from singular higher-order differential equations using boundary triples.

Findings

Derived Weyl m-functions for several boundary conditions.

Constructed regularized quasi-derivatives via Gram-Schmidt process.

Presented explicit matrix-valued Nevanlinna–Herglotz functions.

Abstract

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 boundary 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 a previous paper, 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.