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As an application of a general left-definite spectral theory, Everitt,Littlejohn and Wellman, in 2002, developed the left-definite theory associatedwith 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 explicitlydetermined the domain $\mathcal{D}(A^{2})$ of the self-adjoint operator$A^{2}.$ However, this domain, in their characterization, does not containboundary conditions. In fact, this is a general feature of the left-definiteapproach developed by Littlejohn and Wellman. Yet, the square of thesecond-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 boundaryconditions. In this paper, we show that this domain can, in fact, be expressedusing four separated boundary conditions using the classical GKN(Glazman-Krein-Naimark) theory. In addition, we determine a newcharacterization 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 theGKN theory.

Glazman-Krein-Naimark Theory, Left-Definite Theory and the Square of the Legendre Polynomials Differential Operator