# Multi-Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra

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*W. N. Everitt; L. Markus*

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.

#### Table of Contents

# Table of Contents

## Multi-Interval Linear Ordinary Boundary Value Problems and Complex Symplectic Algebra

- Contents vii8 free
- Multi–interval linear ordinary boundary value problems and complex symplectic algebra 110 free
- 1. Introduction: Goals, Organization 110
- 2. Some definitions for multi–interval systems 211
- 3. Complex symplectic spaces 615
- 4. Single interval quasi–differential systems 1120
- 5. Multi–interval quasi–differential systems 1322
- 6. Boundary symplectic spaces for multi–interval systems 1928
- 7. Finite multi–interval systems 3241
- 8. Examples of complete Lagrangians 4554
- Bibliography 6372