eBook ISBN:  9781470403089 
Product Code:  MEMO/151/715.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $29.40 
eBook ISBN:  9781470403089 
Product Code:  MEMO/151/715.E 
List Price:  $49.00 
MAA Member Price:  $44.10 
AMS Member Price:  $29.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 151; 2001; 64 ppMSC: Primary 34; 51
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.
ReadershipGraduate students and research mathematicians interested in ordinary differential equations and geometry.

Table of Contents

Chapters

1. Introduction: Goals, organization

2. Some definitions for multiinterval systems

3. Complex symplectic spaces

4. Single interval quasidifferential systems

5. Multiinterval quasidifferential systems

6. Boundary symplectic spaces for multiinterval systems

7. Finite multiinterval systems

8. Examples of complete Lagrangians


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

Chapters

1. Introduction: Goals, organization

2. Some definitions for multiinterval systems

3. Complex symplectic spaces

4. Single interval quasidifferential systems

5. Multiinterval quasidifferential systems

6. Boundary symplectic spaces for multiinterval systems

7. Finite multiinterval systems

8. Examples of complete Lagrangians