Abstract
We recall a Fredholm criterion for fully elliptic cusp(pseudo)differential operators on a compact manifold with corners ofarbitrary codimension, acting on suitable Sobolev spaces. Then we give aformula for the index in terms of regularized `trace' functionalssimilar to the residue trace of Wodzicki and Guillemin.
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Connes, A. and Moscovici, H.: The local index formula in noncommutative geometry, Geom. Funct. Anal. 5 (1995), 174-243.
Fedosov, B., Schulze, B.-W. and Tarkhanov, N.: The index of elliptic operators on manifolds with conical points, Sel. Math., New Ser. 5 (1999), 467-506.
Guillemin, V.: A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math. 55 (1985), 131-160.
Lauter, R. and Moroianu, S.: On the index formula of Melrose and Nistor, IMAR preprint, March 2000.
Lauter, R. and Moroianu, S.: Fredholm theory for degenerate pseudodifferential operators on manifolds with fibered boundaries, Comm. Partial Differential Equations 26 (2001), 233-283.
Loya, P. A.: Tempered operators and the heat kernel and complex powers of elliptic pseudodifferential operators, Comm. Partial Differential Equations, to appear.
Loya, P. A.: On the b-pseudodifferential calculus on manifolds with corners, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1998.
Mazzeo, R. R. and Melrose, R. B.: Analytic surgery and the eta invariant, Geom. Funct. Anal. 5 (1995), 14-75.
Mazzeo, R. R. and Melrose, R. B.: Pseudodifferential operators on manifolds with fibred boundaries, Asian J. Math. 2 (1998), 833-866.
Melrose, R. B.: Analysis on Manifolds with Corners, in preparation.
Melrose, R. B.: The Atiyah-Patodi-Singer Index Theorem, Res. Notes Math. 4, A. K. Peters, Wellesley, MA, 1993.
Melrose, R. B.: Spectral and scattering theory for the Laplacian on asymptotically Euclidean space, in: M. Ikawa (ed.), Spectral and Scattering Theory, Proceedings of the Taniguchi International Workshop, held in Sanda, November 1992, Lecture Notes Pure Appl. Math. 162, Marcel Dekker, New York, 1994, pp. 85-130.
Melrose, R. B.: The eta invariant and families of pseudodifferential operators, Math. Res. Lett. 2 (1995), 541-561.
Melrose, R. B.: Geometric Scattering Theory. Cambridge University Press, Cambridge, 1995.
Melrose, R. B. and Nistor, V.: Homology of pseudodifferential operators I. Manifolds with boundary, Amer. J. Math., to appear.
Melrose, R. B. and Nistor, V.: K-theory of C*-algebras of b-pseudodifferential operators, Geom. Funct. Anal. 8 (1998), 88-122.
Melrose, R. B. and Piazza, P.: Analytic K-theory on manifolds with corners, Adv. Math. 92 (1992), 1-26.
Nistor, V.: An index theorem for families of elliptic operators invariant with respect to a bundle of Lie groups, Preprint, 1999.
Radul, A. O.: Lie algebras of differential operators, their central extensions, and W-algebras, Funct. Anal. Appl. 25 (1991), 25-39, 1991. Originally published in Russian: Funktsional. Anal. i Prilozhen. 25 (1991), 33-49.
Schulze, B.-W., Sternin, B. and Shatalov, V.: On the index of differential operators on manifolds with conical singularities, Ann. Global Anal. Geom. 16 (1998), 141-172.
Seeley, R. T.: Complex Powers of an Elliptic Operator, Proc. Symp. Pure Math.-Singular Integrals, Vol. X, Amer. Math. Soc., Providence, RI, 1967, pp. 288-307.
Wodzicki, M.: Noncommutative residue. I. Fundamentals, in: Yu. I. Manin (ed.), K-Theory, Arithmetic and Geometry (Moscow, 1984-1986), Springer-Verlag, Berlin, 1987, pp. 320-399.
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Lauter, R., Moroianu, S. The Index of Cusp Operators on Manifolds with Corners. Annals of Global Analysis and Geometry 21, 31–49 (2002). https://doi.org/10.1023/A:1014283604496
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DOI: https://doi.org/10.1023/A:1014283604496