GlazmanKreinNaimark Theory, LeftDefinite Theory and the Square of the Legendre Polynomials Differential Operator
Abstract
As an application of a general leftdefinite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the leftdefinite theory associated with the classical Legendre selfadjoint secondorder differential operator $A$ in $L^{2}(1,1)$ which has the Legendre polynomials $\{P_{n}% \}_{n=0}^{\infty}$ as eigenfunctions. As a consequence, they explicitly determined the domain $\mathcal{D}(A^{2})$ of the selfadjoint operator $A^{2}.$ However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the leftdefinite approach developed by Littlejohn and Wellman. Yet, the square of the secondorder Legendre expression is in the limit4 case at each end point $x=\pm1$ in $L^{2}(1,1)$ so $\mathcal{D}(A^{2})$ 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 (GlazmanKreinNaimark) theory. In addition, we determine a new characterization of $\mathcal{D}(A^{2})$ that involves four \textit{nonGKN} boundary conditions. These new boundary conditions are surprisingly simple  and natural  and are equivalent to the boundary conditions obtained from the GKN theory.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.07337
 Bibcode:
 2015arXiv151007337L
 Keywords:

 Mathematics  Spectral Theory;
 [2000]Primary 33C45;
 34B24;
 Secondary 34B30