By W. N. Everitt

This research introduces a brand new description and type for the set of all self-adjoint operators (not simply these outlined via differential boundary stipulations) that are generated through a linear elliptic partial differential expression $A(\mathbf{x},D)=\sum_{0\,\leq\,\lefts\right\,\leq\,2m}a_{s} (\mathbf{x})D^{s}\;\text{for all}\;\mathbf{x}\in\Omega$ in a sector $\Omega$, with compact closure $\overline{\Omega}$ and $C^{\infty}$-smooth boundary $\partial\Omega$, in Euclidean area $\mathbb{E}^{r}$ $(r\geq2).$ The order $2m\geq2$ and the spatial measurement $r\geq2$ are arbitrary. We suppose that the coefficients $a_{s}\in C^{\infty}(\overline {\Omega})$ are complex-valued, other than actual for the top order phrases (where $\lefts\right =2m$) which fulfill the uniform ellipticity situation in $\overline{\Omega}$.In addition, $A(\cdot,D)$ is Lagrange symmetric in order that the corresponding linear operator $A$, on its classical area $D(A):=C_{0}^{\infty}(\Omega)\subset L_{2}(\Omega)$, is symmetric; for instance the time-honored Laplacian $\Delta$ and the better order polyharmonic operators $\Delta^{m}$. throughout the equipment of advanced symplectic algebra, which the authors have formerly constructed for usual differential operators, the Stone-von Neumann concept of symmetric linear operators in Hilbert house is reformulated and tailored to the choice of all self-adjoint extensions of $A$ on $D(A)$, by way of an summary generalization of the Glazman-Krein-Naimark (GKN) Theorem.In specific the authors build a common bijective correspondence among the set $\{T\}$ of all such self-adjoint operators on domain names $D(T)\supset D(A)$, and the set $\{\mathsf{L}\}$ of all entire Lagrangian subspaces of the boundary complicated symplectic area $\mathsf{S}=D(T_{1}\,/\,D(T_{0})$, the place $T_{0}$ on $D(T_{0})$ and $T_{1}$ on $D(T_{1})$ are the minimum and maximal operators, respectively, made up our minds by way of $A$ on $D(A)\subset L_{2}(\Omega)$. on the subject of the elliptic partial differential operator $A$, we determine $D(T_{0})=\overset{\text{o}}{W}{}^{2m}(\Omega)$ and supply a singular definition and structural research for $D(T_{1})=\overset{A}{W}{}^{2m}(\Omega)$, which extends the GKN-theory from traditional differential operators to a definite type of elliptic partial differential operators.Thus the boundary complicated symplectic house $\mathsf{S}=\overset{A} {W} {}^{2m}(\Omega)\,/\,\overset{\text{o}}{W}{}^{2m}(\Omega)$ results a category of all self-adjoint extensions of $A$ on $D(A)$, together with these operators that aren't laid out in differential boundary stipulations, yet as an alternative by means of worldwide (i. e. non-local) generalized boundary stipulations. The scope of the speculation is illustrated through a number of favourite, and different relatively strange, self-adjoint operators defined in targeted examples. An Appendix is hooked up to offer the elemental definitions and ideas of differential topology and sensible research on differentiable manifolds. during this Appendix care is taken to checklist and clarify all designated mathematical phrases and emblems - particularly, the notations for Sobolev Hilbert areas and the precise hint theorems. An Acknowledgment and topic Index whole this memoir.