A multi-interval quasi-differential system ({I_{r},M_{r},w_{r}:rinOmega}) consists of a collection of real intervals, ({I_{r}}), as indexed by a finite, or possibly infinite index set (Omega) (where (mathrm{card} (Omega)geqaleph_{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 (rinOmega) 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.

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