Abstract
We prove a general theorem on the behavior of the relative index under surgery for a wide class of Fredholm operators, including relative index theorems for elliptic operators due to Gromov-Lawson, Anghel, Teleman, Booß-Bavnbek-Wojciechowski, et al. as special cases. In conjunction with additional conditions (like symmetry conditions), this theorem permits one to compute the analytical index of a given operator. In particular, we obtain new index formulas for elliptic pseudodifferential operators and quantized canonical transformations on manifolds with conical singularities.
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References
Agranovich, M. S., Dynin, A. S., General elliptic boundary value problems for elliptic systems in higher-dimensional domains, Dokl. Akad. Nauk SSSR, 146:511–514, 1962.
Agranovich, M. S., Elliptic singular integro-differential operators, Uspekhi Matem. Nauk, 20(5):3–20, 1965.
Agranovich, M. S.,. Elliptic boundary problems, in Partial Differential Equations IX. Elliptic Boundary Value Problems, Vol. 79 in Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin-Heidelberg, 1997, pp. 1–144.
Anghel, N., An abstract index theorem on non-compact Riemannian manifolds, Houston J. of Math., 19:223–237, 1993.
Atiyah, M., Singer, I., The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc., 69:422–433, 1963.
Atiyah, M., Global theory of elliptic operators, Proc. of the Int. Symposium on Functional Analysis,21–30, 1969, University of Tokyo Press,Tokyo
Bunke, U., Relative index theory, J. Funct. Anal., 105:63–76, 1992.
Booß-Bavnbek, B., Wojciechowski, K., Elliptic Boundary Problems for Dirac OperatorsBirkhäuser, Boston-Basel-Berlin, 1993.
Dezin, A. A., Invariant Differential Operators and Boundary Value Problems, American Mathematical Society, 1964.
Donnelly, H., Essential spectrum and the heat kernel, J. Funct. Anal., 75:362–381, 1987.
Dynin, A. S., Multidimensional elliptic boundary problems with one unknown function, Dokl. Akad. Nauk SSSR, 141:285–287, 1961.
Epstein, C., Melrose, R., Contact degree and the index of Fourier integral operators, Math. Res. Lett., 5(3):363–381, 1998.
Fedosov, B. V., Schulze, B. W., Tarkhanov, N. N., A Remark on the Index of Symmetric Operators, Preprint 98/4, Univ. Potsdam, Institut für Mathematik, Potsdam, 1998.
Fedosov, B. V., Schulze, B. W., Tarkhanov, N. N., The index of higher order operators on singular surfaces, Pacific J. of Math., 191(1):25–48, 1999.
Gilkey, P. B., Invariance Theory, the Heat Equation and the Atiyah-Singer Index TheoremPublish or Perish, Wilmington, 1984.
Gromov, M., Lawson Jr., H. B., Positive scalar curvature and the Dirac opera-tor on complete Riemannian manifolds, Publ. Math. IHES, 58:295–408, 1983.
Gilkey, P. B., Smith, L., The eta invariant for a class of elliptic boundary value problems, Comm Pure Appl. Math., 36:85–132, 1983.
P. B. Gilkey and L. Smith. The twisted index problem for manifolds with boundary, J. Diff. Geometry, 18(3):393–44, 1983.
Hsiung, C.-C., The signature and G-signature of manifolds with boundary, J.Diff. Geometry, 6:595–598, 1972.
Hsiung, C.-C. A remark on cobordism of manifolds with boundary, Arch. Math., 27:551–555, 1976.
Leichtnam, E., Nest, R., Tsygan, B., Local formula for the index of a Fourier integral operator, Preprint Math. DG/0004022, 2000.
Nazaikinskii, V., Sternin, B., Surgery and the Relative Index of Elliptic Oper-ators, Universität Potsdam, Institut für Mathematik, Potsdam, 1999.
Nazaikinskii, V., Sternin, B., Localization and surgery in index theory of elliptic operators, Dokl. Ross. Akad. Nauk, 370(1):19–23, 2000.
Nazaikinskii, V., Sternin, B., The index locality principle in elliptic theory, Funktsional’nyi Analiz i Prilozhen., 2001 (to appear).
Nazaikinskii, V., Schulze, B.-W., Sternin, B., The Index of Quantized Contact Transformations on Manifolds with Conical Singularities, Preprint 98/16, Univ. Potsdam, Institut für Mathematik, Potsdam, 1998.
Nazaikinskii, V., Schulze, B.-W., Sternin, B., The index of quantized contact transformations on manifolds with conical singularities, Dokl. Ross. Akad. Nauk, 368(5):598–600, 1999.
Schulze, B.-W., Sternin, B., Shatalov, V., On the index of differential operators on manifolds with conical singularities, Annals of Global Analysis and Geometry, 16(2)141–172, 1998.
Stong, R. E., Manifolds with reflecting boundary, J. Duff. Geometry, 9:465–474,1974.
Teleman, N., The index of signature operators on Lipschitz manifolds, Publ.Math. IHES, 58:39–78, 1984.
Weinstein, A., Fourier integral operators, quantization, and the spectrum of a Riemannian manifold, Géométrie Symplectique et Physique Mathématique, 237:289–298, 1976.
Weinstein, A. Some questions about the index of quantized contact transformations, RIMS Kôkûryuku, 104:1–14, 1977.
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Nazaikinskii, V., Sternin, B. (2001). Surgery and the Relative Index in Elliptic Theory. In: Demuth, M., Schulze, BW. (eds) Partial Differential Equations and Spectral Theory. Operator Theory: Advances and Applications, vol 126. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8231-6_26
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DOI: https://doi.org/10.1007/978-3-0348-8231-6_26
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