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 $L^{2}(-1,1)$ 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 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 $x=\pm1$ 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 (Glazman-Krein-Naimark) theory. In addition, we determine a new characterization of $\mathcal{D}(A^{2})$ that involves four \textit{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.