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.