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A multi-interval quasi-differential system $\{I_{r},M_{r},w_{r}:r\in\Omega\}$ consists of a collection of real intervals, $\{I_{r}\}$, as indexed by a finite, or possibly infinite index set $\Omega$ (where $\mathrm{card} (\Omega)\geq\aleph_{0}$ is permissible), on which are assigned ordinary or quasi-differential expressions $M_{r}$ generating unbounded operators in the Hilbert function spaces $L_{r}^{2}\equiv L^{2}(I_{r};w_{r})$, where $w_{r}$ are given, non-negative weight functions. For each fixed $r\in\Omega$ assume that $M_{r}$ is Lagrange symmetric (formally self-adjoint) on $I_{r}$ and hence specifies minimal and maximal closed operators $T_{0,r}$ and $T_{1,r}$, respectively, in $L_{r}^{2}$. However the theory does not require that the corresponding deficiency indices $d_{r}^{-}$ and $d_{r}^{+}$ of $T_{0,r}$ are equal (e. g. the symplectic excess $Ex_{r}=d_{r}^{+}-d_{r}^{-}\neq 0$), in which case there will not exist any self-adjoint extensions of $T_{0,r}$ in $L_{r}^{2}$. In this paper a system Hilbert space $\mathbf{H}:=\sum_{r\,\in\,\Omega}\oplus L_{r}^{2}$ is defined (even for non-countable $\Omega$) with corresponding minimal and maximal system operators $\mathbf{T}_{0}$ and $\mathbf{T}_{1}$ in $\mathbf{H}$. Then the system deficiency indices $\mathbf{d}^{\pm} =\sum_{r\,\in\,\Omega}d_{r}^{\pm}$ are equal (system symplectic excess $Ex=0$), if and only if there exist self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$ in $\mathbf{H}$. The existence is shown of a natural bijective correspondence between the set of all such self-adjoint extensions $\mathbf{T}$ of $\mathbf{T}_{0}$, and the set of all complete Lagrangian subspaces $\mathsf{L}$ of the system boundary complex symplectic space $\mathsf{S}=\mathbf{D(T}_{1})/\mathbf{D(T}_{0})$. This result generalizes the earlier symplectic version of the celebrated GKN-Theorem for single interval systems to multi-interval systems. Examples of such complete Lagrangians, for both finite and infinite dimensional complex symplectic $\mathsf{S}$, illuminate new phenoma for the boundary value problems of multi-interval systems. These concepts have applications to many-particle systems of quantum mechanics, and to other physical problems.
This investigation introduces a new description and classification for the set of all self-adjoint operators (not just those defined by differential boundary conditions) which are generated by a linear elliptic partial differential expression $$A(\mathbf{x},D)=\sum_{0\,\leq\,\left s\right \,\leq\,2m}a_{s} (\mathbf{x})D^{s}\text{ for all }\mathbf{x}\in\Omega$$ in a region $\Omega$, with compact closure $\overline{\Omega}$ and $C^{\infty }$-smooth boundary $\partial\Omega$, in Euclidean space $\mathbb{E}^{r}$ $(r\geq2).$ The order $2m\geq2$ and the spatial dimension $r\geq2$ are arbitrary. We assume that the coefficients $a_{s}\in C^{\infty}(\overline {\Omega})$ are complex-valued, except real for the highest order terms (where $\left s\right =2m$) which satisfy the uniform ellipticity condition in $\overline{\Omega}$. In addition, $A(\cdot,D)$ is Lagrange symmetric so that the corresponding linear operator $A$, on its classical domain $D(A):=C_{0}^{\infty}(\Omega)\subset L_{2}(\Omega)$, is symmetric; for example the familiar Laplacian $\Delta$ and the higher order polyharmonic operators $\Delta^{m}$. Through the methods of complex symplectic algebra, which the authors have previously developed for ordinary differential operators, the Stone-von Neumann theory of symmetric linear operators in Hilbert space is reformulated and adapted to the determination of all self-adjoint extensions of $A$ on $D(A)$, by means of an abstract generalization of the Glazman-Krein-Naimark (GKN) Theorem. In particular the authors construct a natural bijective correspondence between the set $\{T\}$ of all such self-adjoint operators on domains $D(T)\supset D(A)$, and the set $\{\mathsf{L}\}$ of all complete Lagrangian subspaces of the boundary complex symplectic space $\mathsf{S}=D(T_{1})\,/\,D(T_{0})$, where $T_{0}$ on $D(T_{0})$ and $T_{1}$ on $D(T_{1})$ are the minimal and maximal operators, respectively, determined by $A$ on $D(A)\subset L_{2}(\Omega)$. In the case of the elliptic partial differential operator $A$, we verify $D(T_{0})=\overset{\text{o}}{W}{}^{2m}(\Omega)$ and provide a novel definition and structural analysis for $D(T_{1})=\overset{A}{W}{}^{2m}(\Omega)$, which extends the GKN-theory from ordinary differential operators to a certain class of elliptic partial differential operators. Thus the boundary complex symplectic space $\mathsf{S}= \overset{A}{W}{}^{2m}(\Omega)\,/\,\overset{\text{o}}{W}{}^{2m}(\Omega)$ effects a classification of all self-adjoint extensions of $A$ on $D(A)$, including those operators that are not specified by differential boundary conditions, but instead by global (i.e. non-local) generalized boundary conditions. The scope of the theory is illustrated by several familiar, and other quite unusual, self-adjoint operators described in special examples. An Appendix is attached to present the basic definitions and concepts of differential topology and functional analysis on differentiable manifolds. In this Appendix care is taken to list and explain all special mathematical terms and symbols - in particular, the notations for Sobolev Hilbert spaces and the appropriate trace theorems. An Acknowledgment and subject Index complete this Memoir.
In the classical theory of self-adjoint boundary value problems for linear ordinary differential operators there is a fundamental, but rather mysterious, interplay between the symmetric (conjugate) bilinear scalar product of the basic Hilbert space and the skew-symmetric boundary form of the associated differential expression. This book presents a new conceptual framework, leading to an effective structured method, for analyzing and classifying all such self-adjoint boundary conditions. The program is carried out by introducing innovative new mathematical structures which relate the Hilbert space to a complex symplectic space.This work offers the first systematic detailed treatment in the literature of these two topics: complex symplectic spaces - their geometry and linear algebra - and quasi-differential operators. This title features: authoritative and systematic exposition of the classical theory for self-adjoint linear ordinary differential operators (including a review of all relevant topics in texts of Naimark, and Dunford and Schwartz); introduction and development of new methods of complex symplectic linear algebra and geometry and of quasi-differential operators, offering the only extensive treatment of these topics in book form; new conceptual and structured methods for self-adjoint boundary value problems; and, extensive and exhaustive tabulations of all existing kinds of self-adjoint boundary conditions for regular and for singular ordinary quasi-differential operators of all orders up through six.