New Titles  |  FAQ  |  Keep Informed  |  Review Cart  |  Contact Us Quick Search (Advanced Search ) Browse by Subject General Interest Logic & Foundations Number Theory Algebra & Algebraic Geometry Discrete Math & Combinatorics Analysis Differential Equations Geometry & Topology Probability & Statistics Applications Mathematical Physics Math Education

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
 SEARCH THIS BOOK:
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

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.