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Complex symplectic geometry with applications to ordinary differential operators
Author(s):
W.
N.
Everitt;
L.
Markus
Journal:
Trans. Amer. Math. Soc.
351
(1999),
4905-4945.
MSC (1991):
Primary 34B05, 34L05;
Secondary 47B25, 58F05
Posted:
July 20, 1999
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Abstract:
Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.
References:
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- Everitt, W.N., On the deficiency index problem for ordinary differential operators 1910-1977, Proceedings of The 1977 Uppsala International Conference: Differential Equations, 62-81, Published by the University of Uppsala, Sweden, 1977, distributed by Almquist and Wiksell International Stockholm, Sweden, pp. 62-81. MR 57:16788
- [EV]
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- [EM]
- Everitt, W.N. and Markus, L., Boundary Value Problems and Symplectic Algebra for Ordinary Differential and Quasi-Differential Operators, Math. Surveys and Monographs, vol. 61, Amer. Math. Soc., Providence, RI, 1999. CMP 99:03
- [EM1]
- -, The Glazman-Krein-Naimark theorem for ordinary differential operators, in New Results on Operator Theory and Its Applications: The I. M. Glazman Memorial Volume, Operator Theory: Advances and Applications, vol. 98, Birkhäuser, Basel, 1997, pp. 118-130. MR 99c:47070
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- Glazman, I.M., On the theory of singular differential operators, Uspehi Math. Nauk 40 (1950), 102-135; English translation in Amer. Math. Soc. Translations (1) 4 (1962), 331-372. MR 13:254d; MR 15:327a
- [MA]
- Markus, L., Hamiltonian dynamics and symplectic manifolds, Lecture Notes, University of Minnesota, University of Minnesota Bookstores, 1973, 1-256.
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- Naimark, M.A., Linear differential operators: Part II, Ungar, New York, 1968; translated from the manuscript of the second Russian edition, ``Nauka'', Moscow, 1969. MR 41:7185
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Additional Information:
W.
N.
Everitt
Affiliation:
Department of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, England, United Kingdom
Email:
w.n.everitt@bham.ac.uk
L.
Markus
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
markus@math.umn.edu
DOI:
10.1090/S0002-9947-99-02418-6
PII:
S 0002-9947(99)02418-6
Keywords:
Ordinary linear differential operators,
deficiency indices,
symmetric boundary conditions,
symplectic geometry
Received by editor(s):
August 19, 1997
Posted:
July 20, 1999
Dedicated:
Dedicated to Professor Hugh L. Turrittin on the occasion of his ninetieth birthday
Copyright of article:
Copyright
1999,
American Mathematical Society
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