Memoirs of the American Mathematical Society 2001; 64 pp; softcover Volume: 151 ISBN10: 0821826697 ISBN13: 9780821826690 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/151/715
 A multiinterval quasidifferential 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 quasidifferential 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, nonnegative weight functions. For each fixed \(r\in\Omega\) assume that \(M_{r}\) is Lagrange symmetric (formally selfadjoint) 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 selfadjoint 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 noncountable \(\Omega\)) 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 selfadjoint 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 selfadjoint 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 GKNTheorem for single interval systems to multiinterval 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 multiinterval systems. These concepts have applications to manyparticle systems of quantum mechanics, and to other physical problems. Readership Graduate students and research mathematicians interested in ordinary differential equations and geometry. Table of Contents  Introduction: Goals, organization
 Some definitions for multiinterval systems
 Complex symplectic spaces
 Single interval quasidifferential systems
 Multiinterval quasidifferential systems
 Boundary symplectic spaces for multiinterval systems
 Finite multiinterval systems
 Examples of complete Lagrangians
 Bibliography
