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Boundary triples and Weyl \(m\)-functions for powers of the Jacobi differential operator. (English) Zbl 07216743

In this article, 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 the first explicit examples to stem from singular higher-order differential equations.
The definition of the boundary triples involves taking pieces, determined in [D. Frymark and C. Liaw, J. Math. Anal. Appl. 489, No. 1, Article ID 124155, 31 p. (2020; Zbl 07210874)], of the principal and non-principal solutions of the differential equation. These pieces then are put 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.

MSC:

34B20 Weyl theory and its generalizations for ordinary differential equations
34B24 Sturm-Liouville theory
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
47B25 Linear symmetric and selfadjoint operators (unbounded)
47E05 General theory of ordinary differential operators

Citations:

Zbl 07210874

Software:

SLEIGN2; DLMF
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References:

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