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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:444

Article

Littlejohn, Lance L.¹  Wicks, Quinn¹ 

[1] Baylor Univ, Dept Math, One Bear Pl 97328, Waco, TX 76798 USA

关键词: Legendre polynomials;    Self-adjoint operator;    Left-definite theory;    Glazman-Krein-Naimark theory;    Boundary conditions;   

DOI  :  10.1016/j.jmaa.2016.06.010

来源: Elsevier

【 摘 要 】

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 L-2(-1, 1) which has the Legendre polynomials {Pn}(n=0)(infinity) as eigenfunctions. As a consequence, they explicitly determined the domain D(A(2)) of the self-adjoint operator A(2). However, this domain, in their characterization, does not contain boundary conditions. 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 end point a = 11 in L-2(-1, 1) so D(A(2)) should exhibit four boundary conditions. In this paper, 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(A(2)) 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

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