Glazman-Krein-Naimark theory, left-definite theory, and the square of the Legendre polynomials differential operator.

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L² (−1, 1) which has the Legendre polynomials {Pn} ∞ n=0 as eigenfunctions. As a consequence, they explicitly determined the domain D(A2) of the self-adjoint operator A2 . However, this domain, in their characterization, does not contain boundary conditions in its formulation. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each endpoint x = ±1 in L2 (−1, 1), so D(A2) should exhibit four boundary conditions. In this thesis, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman-Krein-Naimark) theory. In addition, we determine a new characterization of D(A2) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple and natural, and are equivalent to the boundary conditions obtained from the GKN theory.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Doctoral

Grantor

Baylor University.

Year dc:date.issued

2016

Author dc:creator

• Wicks, Quinn Callahan, 1984-

Advisor dc:contributor.advisor

• Littlejohn, Lance, 1951-

• Left-definite Theory.
• Glazman-Krein-Naimark theory.
• Legendre differential equation.

Statement dc:rights

• Baylor University works are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. Contact libraryquestions@baylor.edu for inquiries about permission.

Language dc:language.iso

en

Handle dc:identifier.uri

https://hdl.handle.net/2104/9654

OAI identifier oai:identifier

oai:baylor-ir.tdl.org:2104/9654