Hörmander index in finite-dimensional case

doi:10.1007/s11464-018-0702-3

Front. Math. China

2018, Vol. 13

Issue (3) : 725-761 https://doi.org/10.1007/s11464-018-0702-3

RESEARCH ARTICLE

Hörmander index in finite-dimensional case

Yuting ZHOU¹, Li WU², Chaofeng ZHU¹(

)

¹. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
². Department of Mathematics, Shandong University, Jinan 250100, China

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Abstract

We calculate the Hörmander index in the finite-dimensional case. Then we use the result to give some iteration inequalities, and prove almost existence of mean indices for given complete autonomous Hamiltonian system on compact symplectic manifold with symplectic trivial tangent bundle and given autonomous Hamiltonian system on regular compact energy hypersurface of symplectic manifold with symplectic trivial tangent bundle.

Keywords Maslov index Hörmander index Maslov-type index symplectic reduction

Corresponding Author(s): Chaofeng ZHU

Issue Date: 11 June 2018

Cite this article:

Yuting ZHOU,Li WU,Chaofeng ZHU. Hörmander index in finite-dimensional case[J]. Front. Math. China, 2018, 13(3): 725-761.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-018-0702-3
https://academic.hep.com.cn/fmc/EN/Y2018/V13/I3/725

1	Booss-Bavnbek B, Furutani K. Symplectic functional analysis and spectral invariants. In: Booss-Bavnbek B, Wojciechowski K, eds. Geometric Aspects of Partial Differential Equations. Contemp Math, Vol 242. Providence: Amer Math Soc, 1999, 53–83 https://doi.org/10.1090/conm/242/03661
2	Booß-Bavnbek B, Zhu C. The Maslov index in weak symplectic functional analysis. Ann Global Anal Geom, 2013, 44: 283–318 https://doi.org/10.1007/s10455-013-9367-z
3	Booß-Bavnbek B, Zhu C. Maslov Index in Symplectic Banach Spaces. Mem Amer Math Soc (to appear), arXiv: 1406.1569v4 https://doi.org/10.1090/memo/1201
4	de Gosson M. On the usefulness of an index due to Leray for studying the intersections of Lagrangian and symplectic paths. J Math Pures Appl (9), 2009, 91(6): 598–613
5	Duistermaat J J. On the Morse index in variational calculus. Adv Math, 1976, 21(2): 173–195 https://doi.org/10.1016/0001-8708(76)90074-8
6	Ekeland I. Convexity Methods in Hamiltonian Mechanics. Ergeb Math Grenzgeb (3), Vol 19. Berlin: Springer-Verlag, 1990 https://doi.org/10.1007/978-3-642-74331-3
7	Hofer H, Zehnder E. Symplectic Invariants and Hamiltonian Dynamics. Modern Birkhäuser Classics. Basel: Birkhäuser Verlag, 2011 https://doi.org/10.1007/978-3-0348-0104-1
8	Hörmander L. Fourier integral operators. I. Acta Math, 1971, 127(1-2): 79–183 https://doi.org/10.1007/BF02392052
9	Kato T. Perturbation Theory for Linear Operators. Berlin: Springer, 1995 https://doi.org/10.1007/978-3-642-66282-9
10	Kingman J F C. The ergodic theory of subadditive stochastic processes. J Roy Stat Soc Ser B, 1968, 30: 499–510
11	Lee J M. Introduction to Smooth Manifolds. Grad Texts in Math, Vol 218. New York: Springer-Verlag, 2003 https://doi.org/10.1007/978-0-387-21752-9
12	Liu C. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete Contin Dyn Syst, 2010, 27(1): 337–355 https://doi.org/10.3934/dcds.2010.27.337
13	Liu C, Zhang D. Iteration theory of L-index and multiplicity of brake orbits. J Differential Equations, 2014, 257(4): 1194–1245 https://doi.org/10.1016/j.jde.2014.05.006
14	Liu C, Zhang D. Seifert conjecture in the even convex case. Commun Pure Appl Math, 2014, 67(10): 1563–1604 https://doi.org/10.1002/cpa.21525
15	Long Y. The minimal period problem of classical Hamiltonian systems with even potentials. Ann Inst H Poincaré Anal Non Linéaire, 1993, 10(6): 605–626 https://doi.org/10.1016/S0294-1449(16)30199-8
16	Long Y. Index Theory for Symplectic Paths with Applications. Progr Math, Vol 27. Basel: Birkhäuser Verlag, 2002 https://doi.org/10.1007/978-3-0348-8175-3
17	Long Y, Zhang D, Zhu C. Multiple brake orbits in bounded convex symmetric domains. Adv Math, 2006, 203(2): 568–635 https://doi.org/10.1016/j.aim.2005.05.005
18	Salamon D, Zehnder E. Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun Pure Appl Math, 1992, 45(10): 1303–1360 https://doi.org/10.1002/cpa.3160451004
19	Zhu C. A generalized Morse index theorem. In: Booβ-Bavnbek B, Klimek S, Lesch M, Zhang W P, eds. Analysis, Geometry and Topology of Elliptic Operators. Hackensack:World Sci Publ, 2006, 493–540 https://doi.org/10.1142/9789812773609_0020

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