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.