2015/10/26 by Lance L. Littlejohn, Quinn Wicks, Littlejohn, Lance L. +1
Mathematics · #34B24 #FOS: Mathematics #Secondary 34B30 #Spectral Theory (math.SP) #[2000]Primary 33C45 #math.SP #msc:33C45 #msc:34B24 #msc:34B30
paper · pdf · doi:10.48550/arxiv.1510.07337
arxiv created 2015/10/26 · arxiv updated 2015/10/27
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 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 x=±1 in L2(-1,1) so D(A2) 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(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.