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2005 | 3 | 3 | 558-577
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General spectral flow formula for fixed maximal domain

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Języki publikacji
EN
Abstrakty
EN
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.
Wydawca
Czasopismo
Rocznik
Tom
3
Numer
3
Strony
558-577
Opis fizyczny
Daty
wydano
2005-09-01
online
2005-09-01
Twórcy
autor
Bibliografia
  • [1] W. Ambrose: “The index theorem in Riemannian geometry”, Ann. of Math., Vol. 73, (1961), pp. 49–86. http://dx.doi.org/10.2307/1970282
  • [2] V.I. Arnol'd: “Characteristic class entering in quantization conditions”, Funkcional. Anal. i Priložen., Vol. 1, (1967), pp. 1–14 (in russian); Functional Anal. Appl., Vol. 1, (1967), pp. 1–13 (in english); Théorie des perturbations et méthodes asymptotiques, Dunod, Gauthier-Villars, Paris, 1972, pp. 341–361 (in french). http://dx.doi.org/10.1007/BF01075861
  • [3] M.F. Atiyah, V.K. Patodi and I.M. Singer: “Spectral asymmetry and Riemannian geometry, I”, Math. Proc. Cambridge Phil. Soc., Vol. 77, (1975), pp. 43–69. http://dx.doi.org/10.1017/S0305004100049410
  • [4] B. Bojarski: “The abstract linear conjugation problem and Fredholm pairs of subspaces” In: In Memoriam I.N. Vekua, Tbilisi Univ., Tbilisi, 1979, pp. 45–60 (in russian).
  • [5] B. Booss-Bavnbek and K. Furutani: “The Maslov index-a functional analytical definition and the spectral flow formula”, Tokyo. J. Math., Vol. 21, (1998), pp. 1–34. http://dx.doi.org/10.3836/tjm/1270041982
  • [6] B. Booss-Bavnbek, K. Furutani and N. Otsuki: “Criss-cross reduction of the Maslov index and a proof of the Yoshida-Nicolaescu Theorem”, Tokyo J. Math., Vol. 24, (2001), pp. 113–128. http://dx.doi.org/10.3836/tjm/1255958316
  • [7] B. Booss-Bavnbek, M. Lesch and J. Phillips: “Unbounded Fredholm operators and spectral flow”, Canad. J. Math., Vol. 57(2), (2005), pp. 225–250, arXiv: math.FA/0108014.
  • [8] B. Booss-Bavnbek, M. Lesch and C. Zhu: “Elliptic differential operators on compact manifolds with smooth boundary”, in preparation.
  • [9] B. Booss-Bavnbek, M. Marcolli and B.-L. Wang: “Weak UCP and perturbed monopole equations”, Internat. J. Math., Vol. 13(9), (2002), pp. 987–1008. http://dx.doi.org/10.1142/S0129167X02001551
  • [10] B. Booss-Bavnbek and K.P. Wojciechowski: Elliptic Boundary Problems for Dirac Operators, Birkhäuser, Boston, 1993.
  • [11] B. Booss-Bavnbek and C. Zhu: Weak Symplectic Functional Analysis and General Spectral Flow Formula, Preprint, Roskilde, December 2003, arXiv: math.DG/0406139.
  • [12] S.E. Cappell, R. Lee and E.Y. Miller: “Selfadjoint elliptic operators and manifold decompositions Part II: Spectral flow and Maslov index”, Comm. Pure Appl. Math., Vol. 49, (1996), pp. 869–909. http://dx.doi.org/10.1002/(SICI)1097-0312(199609)49:9<869::AID-CPA1>3.0.CO;2-5
  • [13] A. Carey and J. Phillips: Spectral Flow in Fredholm Modules, Eta Invariants and the JLO Cocycle, Preprint 2003, arXiv: math.KT/0308161.
  • [14] J.J. Duistermaat: “On the Morse index in variational calculus”, Adv. Math., Vol. 21, (1976), pp. 173–195. http://dx.doi.org/10.1016/0001-8708(76)90074-8
  • [15] A. Floer: “A relative Morse index for the symplectic action”, Comm. Pure Appl. Math., Vol. 41, (1988), pp. 393–407.
  • [16] B. Himpel, P. Kirk and M. Lesch: “Calderón projector for the Hessian of the Chern-Simons function on a 3-manifold with boundary”, Proc. London. Math. Soc., Vol. 89, (2004), pp. 241–272, arXiv: math.GT/0302234. http://dx.doi.org/10.1112/S0024611504014728
  • [17] T. Kato: Perturbation Theory for Linear Operators, 2nd ed., Springer-Verlag, Berlin, 1966; 1976 corrected printing; 1980.
  • [18] P. Kirk and M. Lesch: “The η-invariant, Maslov index, and spectral flow for Diractype operators on manifolds with boundary”, Forum Math., Vol. 16(4), (2004), pp. 553–629, arXiv: math.DG/0012123. http://dx.doi.org/10.1515/form.2004.027
  • [19] B. Lawruk, J. Śniatycki and W.M. Tulczyjew: “Special symplectic spaces”, J. Differential Equations, Vol. 17, (1975), pp. 477–497. http://dx.doi.org/10.1016/0022-0396(75)90057-1
  • [20] Y. Long and C. Zhu: “Maslov-type index theory for symplectic paths and spectral flow (II)”, Chinese Ann. of Math. B, Vol. 21(1), (2000), pp. 89–108. http://dx.doi.org/10.1142/S0252959900000133
  • [21] M. Morse: The Calculus of Variations in the Large, Vol. 18, A.M.S. Coll. Publ., Amer. Math. Soc., New York, 1934.
  • [22] L. Nicolaescu: “The Maslov index, the spectral flow, and decomposition of manifolds”, Duke Math. J., Vol. 80, (1995), pp. 485–533. http://dx.doi.org/10.1215/S0012-7094-95-08018-1
  • [23] J. Phillips: “Self-adjoint Fredholm operators and spectral flow”, Canad. Math. Bull., Vol. 39, (1996), pp. 460–467.
  • [24] P. Piccione and D.V. Tausk: “The Maslov index and a generalized Morse index theorem for non-positive definite metrics”, C. R. Acad. Sci. Paris Sér. I. Math., Vol. 331, (2000), pp. 385–389.
  • [25] “The Morse index theorem in semi-Riemannian Geometry”, Topology, Vol. 41, (2002), pp. 1123–1159, arXiv: math.DG/0011090.
  • [26] A. Pliś: “A smooth linear elliptic differential equation without any solution in a sphere”, Comm. Pure Appl. Math., Vol. 14, (1961), pp. 599–617.
  • [27] J.V. Ralston: “Deficiency indices of symmetric operators with elliptic boundary conditions”, Comm. Pure Appl. Math., Vol. 23, (1970), pp. 221–232.
  • [28] J. Robbin and D. Salamon: “The Maslov index for paths”, Topology, Vol. 32, (1993), pp. 823–844. http://dx.doi.org/10.1016/0040-9383(93)90052-W
  • [29] R.T. Seeley: “Singular integrals and boundary value problems”, Amer. J. Math., Vol. 88, (1966), pp. 781–809. http://dx.doi.org/10.2307/2373078
  • [30] K.P. Wojciechowski: “Spectral flow and the general linear conjugation problem”, Simon Stevin, Vol. 59, (1985), pp. 59–91.
  • [31] Tomoyoshi Yoshida: “Floer homology and splittings of manifolds”, Ann. of Math., Vol. 134, (1991), pp. 277–323. http://dx.doi.org/10.2307/2944348
  • [32] C. Zhu: Maslov-type index theory and closed characteristics on compact convex hypersurfaces in ℝ2n , Thesis (PhD), Nankai Institute, Tianjin, 2000 (in Chinese).
  • [33] C. Zhu: The Morse Index Theorem for Regular Lagrangian Systems, Preprint September 2001 (arXiv: math.DG/0109117) (first version); MPI Preprint 2003, no. 55 (modified version).
  • [34] C. Zhu and Y. Long: “Maslov-type index theory for symplectic paths and spectral flow. (I)”, Chinese Ann. of Math. B, Vol. 20, (1999), pp. 413–424. http://dx.doi.org/10.1142/S0252959999000485
Typ dokumentu
Bibliografia
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