Fredholm-Lagrangian-Grassmannian and the Maslov index

Journal of Geometry and Physics

Volumn 51, Issue 3, 2004, Pages 269-331

  Furutani, Kenro ^a  

^a TOKYO UNIVERSITY OF SCIENCE   (Japan)

Author keywords

Elliptic operator; Fredholm operator; Fredholm pair; Fredholm Lagrangian Grassmannian; H rmander index; K group; Kashiwara index; Lagrangian subspace; Leray index; Maslov index; Symplectic Hilbert space; Unitary group

Indexed keywords

EID: 2542428388     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.geomphys.2004.04.001     Document Type: Review

References (37)

1. Arnold V.I. Characteristic class entering in quantization conditions Funct. Anal. Appl. 1 1967 1-13
2. Atiyah M.F. Patodi V.K. Singer I.M. Spectral asymmetry and Riemannian geometry: I Math. Proc. Camb. Phil. Soc. 77 1975 43-69
3. Atiyah M.F. Patodi V.K. Singer I.M. Spectral asymmetry and Riemannian geometry: II Math. Proc. Camb. Phil. Soc. 78 1975 405-432
4. Atiyah M.F. Patodi V.K. Singer I.M. Spectral asymmetry and Riemannian geometry: III Math. Proc. Camb. Phil. Soc. 79 1976 71-99
5. Atiyah M.F. Singer I.M. Index theory for skew-adjoint Fredholm operators Inst. Hautes Études Sci. Publ. Math. 37 1969 5-26
6. Booss-Bavnbek B. Furutani K. The Maslov index-a functional analytical definition and the spectral flow formula Tokyo J. Math. 21 1998 1-34
7. B. Booss-Bavnbek, K. Furutani, Symplectic functional analysis and spectral invariants, in: B. Booss-Bavnbek, K.P. Wojciechowski (Eds.), Geometric Aspects of Partial Differential Equations, Am. Math. Soc. Series Contemporary Mathematics, vol. 242, Providence, RI, 1999, pp. 53-83.
8. B. Booss-Bavnbek, K. Furutani, N. Otsuki, Criss-cross reduction of the Maslov index and a proof of the Yoshida-Nicolaescu Theorem, Tokyo J. Math. 24 (1) (2001).
9. Cordes H.O. Labrousse J.P. The invariance of the index in the metric of closed operators J. Math. Mech. 12 1963 693-719
10. Cappell S.E. Lee R. Miller E.Y. On the Maslov index Comm. Pure Appl. Math. 47 1994 121-186
11. Cappell S.E. Lee R. Miller E.Y. Selfadjoint elliptic operators and manifold decompositions: Part I. Low eigenmodes and stretching Comm. Pure Appl. Math. 49 1996 825-866
12. Cappell S.E. Lee R. Miller E.Y. Selfadjoint elliptic operators and manifold decompositions: Part II. Spectral flow and Maslov index Comm. Pure Appl. Math. 49 1996 869-909
13. Cappell S.E. Lee R. Miller E.Y. Selfadjoint elliptic operators and manifold decompositions: Part III. Determinant line bundles and Lagrangian intersection Comm. Pure Appl. Math. 52 1999 543-611
14. M. Daniel, P. Kirk, With an appendix by K.P. Wojciechowski: a general splitting formula for the spectral flow, Michigan Math. J. 46 (1999) 589-617.
15. Duistermaat J.J. On the Morse index in variational calculus Adv. Math. 21 1976 173-195
16. Floer A. A relative Morse index for the symplectic action Comm. Pure Appl. Math. 41 1988 393-407
17. Furutani K. Otsuki N. Spectral flow and intersection numbers J. Math. Kyoto Univ. 33 1993 261-283
18. Furutani K. Otsuki N. Maslov index in the infinite dimension and a splitting formula for a spectral flow Jpn. J. Math. 282 2002 215-243
19. Furutani K. On the Quillen determinant J. Geom. Phys. 49 2004 366-375
20. de Gosson M. La définition de l'indice de Maslov sans hypothèse de transversalité CR Acad. Sci. Paris, Série I 310 1990 279-282
21. de Gosson M. The structure of q-symplectic geometry J. Math. Pures Appl. 71 1992 429-453
22. Grubb G. Spectral asymptotics for the soft selfadjoint extension of asymmetric elliptic differential operator J. Oper. Thoery 10 1983 9-20
23. Hoermander L. Pseudo-differential operators and non-elliptic boundary problems Ann. Math. 83 1966 129-209
24. L. Hoermander, Fourier integral operators I, Acta Math. 127 (1-2) (1971) 79-183.
25. L. Hoermander, The Analysis of Linear Partial Differential Operators III, Springer, Berlin, 1985.
26. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1980.
27. Kalton N.J. Swanson R.C. A symplectic Banach space with no Lagrangian subspaces Trans. Am. Math. Soc. 2731 1982 385-392
28. J. Leray, Analyse Lagrangiénne et mécanique quantiqu: Une structure mathématique apparentée aux développements asymptotiques et à l'indice de Maslov, Série Math. Pure et Appl., IRMP, Strasbourg, 1978 (English translation 1981, MIT Press).
29. Nicolaescu L. The Maslov index, the spectral flow, and decomposition of manifolds Duke Math. J. 80 1995 485-533
30. Otsuki N. Furutani K. Spectral flow and Maslow index arising from Lagrangian intersections Tokyo J. Math. 14 1991 135-150
31. Phillips J. Self-adjoint Fredholm operators and spectral flow Can. Math. Bull. 39 1996 460-467
32. Ralston J.V. Deficiency indices of symmetric operators with elliptic boundary conditions Comm. Pure Appl. Math. 23 1970 221-232
33. Robbin J. Salamon D. The Maslov index for paths Topology 32 1993 827-844
34. J.M. Souriau, Construction explicite de l'indice de Maslov, in: Group Theoretical Methods in Physics, Springer Lecture Notes in Physics, vol. 50, Berlin, 1975, pp. 117-148.
35. Swanson R.C. Fredholm intersection theory and elliptic boundary deformation problems: I J. Diff. Eqs. 28 1978 189-201
36. Taubes C.H. Casson's invariant and gauge theory J. Differ. Geom. 31 1990 547-599
37. Yoshida T. Floer homology and splittings of manifolds Ann. Math. 134 1991 277-323