Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complex symplectic geometry with applications
to ordinary differential operators


Authors: W. N. Everitt and L. Markus
Journal: Trans. Amer. Math. Soc. 351 (1999), 4905-4945
MSC (1991): Primary 34B05, 34L05; Secondary 47B25, 58F05
Published electronically: July 20, 1999
MathSciNet review: 1637066
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AG] Akhiezer, N.I. and Glazman, I.M., Theory of linear operators in Hilbert space: volumes I and II, Pitman and Scottish Academic Press, London, 1981; translated from the third Russian edition of 1977. MR 83i:47001
  • [AM] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1978. Second edition, revised and enlarged; With the assistance of Tudor Raţiu and Richard Cushman. MR 515141 (81e:58025)
  • [DS] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745 (32 #6181)
  • [EI] W. N. Everitt, On the deficiency index problem for ordinary differential operators 1910–1976, Differential equations (Proc. Internat. Conf., Uppsala, 1977) Almqvist & Wiksell, Stockholm, 1977, pp. 62–81. Sympos. Univ. Upsaliensis Ann. Quingentesimum Celebrantis, No. 7. MR 0477247 (57 #16788)
  • [EV] -, Linear ordinary quasi-differential expressions, Lecture notes for The Fourth International Symposium on Differential Equations and Differential Geometry, Beijing, Peoples' Republic of China, 1-28. (Department of Mathematics, University of Peking, Peoples' Republic of China; 1986).
  • [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] W. N. Everitt and L. Markus, The Glazman-Krein-Naimark theorem for ordinary differential operators, New results in operator theory and its applications, Oper. Theory Adv. Appl., vol. 98, Birkhäuser, Basel, 1997, pp. 118–130. MR 1478469 (99c:47070)
  • [ER] W. N. Everitt and D. Race, Some remarks on linear ordinary quasidifferential expressions, Proc. London Math. Soc. (3) 54 (1987), no. 2, 300–320. MR 872809 (88b:34014), http://dx.doi.org/10.1112/plms/s3-54.2.300
  • [EZ] W. N. Everitt and A. Zettl, Differential operators generated by a countable number of quasi-differential expressions on the real line, Proc. London Math. Soc. (3) 64 (1992), no. 3, 524–544. MR 1152996 (93k:34182), http://dx.doi.org/10.1112/plms/s3-64.3.524
  • [GZ] I. M. Glazman, On the theory of singular differential operators, Uspehi Matem. Nauk (N.S.) 5 (1950), no. 6(40), 102–135 (Russian). MR 0043389 (13,254d)
    I. M. Glazman, On the theory of singular differential operators, Amer. Math. Soc. Translation 1953 (1953), no. 96, 43. MR 0058132 (15,327a)
  • [MA] Markus, L., Hamiltonian dynamics and symplectic manifolds, Lecture Notes, University of Minnesota, University of Minnesota Bookstores, 1973, 1-256.
  • [MH] Kenneth R. Meyer and Glen R. Hall, Introduction to Hamiltonian dynamical systems and the 𝑁-body problem, Applied Mathematical Sciences, vol. 90, Springer-Verlag, New York, 1992. MR 1140006 (93b:70002)
  • [MS] Dusa McDuff and Dietmar Salamon, Introduction to symplectic topology, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. Oxford Science Publications. MR 1373431 (97b:58062)
  • [NA] J. B. Garner, On the nonsolvability of second order boundary value problems, J. Math. Anal. Appl. 31 (1970), 154–159. MR 0262579 (41 #7185)
  • [TU] H. L. Turrittin, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27–66. MR 0068689 (16,925a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34B05, 34L05, 47B25, 58F05

Retrieve articles in all journals with MSC (1991): 34B05, 34L05, 47B25, 58F05


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: http://dx.doi.org/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
Published electronically: July 20, 1999
Dedicated: Dedicated to Professor Hugh L. Turrittin on the occasion of his ninetieth birthday
Article copyright: © Copyright 1999 American Mathematical Society