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Complex symplectic spaces and boundary value problems


Authors: W. N. Everitt and L. Markus
Journal: Bull. Amer. Math. Soc. 42 (2005), 461-500
MSC (2000): Primary 51A50, 46A03, 37K05; Secondary 46C20, 35J40, 34B05
DOI: https://doi.org/10.1090/S0273-0979-05-01078-5
Published electronically: July 7, 2005
MathSciNet review: 2163706
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Abstract: This paper presents a review and summary of recent research on the boundary value problems for linear ordinary and partial differential equations, with special attention to the investigations of the current authors emphasizing the applications of complex symplectic spaces.

In the first part of the previous century, Stone and von Neumann formulated the theory of self-adjoint extensions of symmetric linear operators on a Hilbert space; in this connection Stone developed the properties of self-adjoint differential operators generated by boundary value problems for linear ordinary differential equations. Later, in diverse papers, Glazman, Krein and Naimark introduced certain algebraic techniques for the treatment of appropriate generalized boundary conditions. During the past dozen years, in a number of monographs and memoirs, the current authors of this expository summary have developed an extensive algebraic structure, complex symplectic spaces, with applications to both ordinary and partial linear boundary value problems.

As a consequence of the use of complex symplectic spaces, the results offer new insights into the theory and use of indefinite inner product spaces, particularly Krein spaces, from an algebraic viewpoint. For instance, detailed information is obtained concerning the separation and coupling of the boundary conditions at the endpoints of the intervals for ordinary differential operators (see the Balanced Intersection Principle), and the introduction of the generalized boundary conditions over the region for some elliptic partial differential operators (see the Harmonic operator).


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Additional Information

W. N. Everitt
Affiliation: School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, UK
Email: w.n.everitt@bham.ac.uk

L. Markus
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA
Email: markus@math.umn.edu

DOI: https://doi.org/10.1090/S0273-0979-05-01078-5
Keywords: Complex symplectic space, indefinite inner product space, topological linear space, ordinary and partial linear boundary value problem
Received by editor(s): March 10, 2005
Published electronically: July 7, 2005
Dedicated: Dedicated toProfessor Jaroslav Kurzweilon the occasion of his 80-th birthday
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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