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Multi-Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra
W. N. Everitt, University of Birmingham, England, and L. Markus, University of Minnesota, Minneapolis, MN

Memoirs of the American Mathematical Society
2001; 64 pp; softcover
Volume: 151
ISBN-10: 0-8218-2669-7
ISBN-13: 978-0-8218-2669-0
List Price: US$49
Individual Members: US$29.40
Institutional Members: US$39.20
Order Code: MEMO/151/715
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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.


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

Table of Contents

  • Introduction: Goals, organization
  • Some definitions for multi-interval systems
  • Complex symplectic spaces
  • Single interval quasi-differential systems
  • Multi-interval quasi-differential systems
  • Boundary symplectic spaces for multi-interval systems
  • Finite multi-interval systems
  • Examples of complete Lagrangians
  • Bibliography
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