Properties and decompositions of domains for powers of the Jacobi differential operator

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:489

Article

Frymark, Dale¹  Liaw, Constanze^2,3 

[1] Stockholm Univ, Dept Math, Kraftriket 6, S-10691 Stockholm, Sweden

[2] Univ Delaware, Dept Math Sci, 501 Ewing Hall, Newark, DE 19716 USA

[3] Baylor Univ, CASPER, One Bear Pl 97328, Waco, TX 76798 USA

关键词: Self-adjoint extension theory;    Sturm-Liouville operators;    Left-definite theory;    Boundary conditions;    Maximal domain;    Minimal domain;   

DOI  :  10.1016/j.jmaa.2020.124155

来源: Elsevier

【 摘 要 】

We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains. The maximal domain for n-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of functions in the underlying Hilbert space can then be classified, for j is an element of N-0, by (1 - x)(j), (1 + x)(j), (1 - x)(-alpha+j) and (1 + x)(beta+j). Most of these behaviors can only occur when functions are in the associated minimal domain, and this leads to a formulation of the defect spaces with a convenient basis. Self-adjoint extensions, including the important left-definite domain, are then given in terms of the new basis functions for the defect spaces using GKN theory. Comments are made for the Laguerre operator as well. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free