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.

Readership

Graduate students and research mathematicians interested in
ordinary differential equations and geometry.