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Resolvents of cone pseudodifferential operators, asymptotic expansions and applications

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Abstract

We study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted Sobolev spaces over a compact manifold with boundary. We obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity, and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula.

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References

  1. Bleistein N. and Handelsman R.A. (1986). Asymptotic expansions of integrals. Dover, New York

    Google Scholar 

  2. Brüning J. and Seeley R. (1987). The resolvent expansion for second order regular singular operators. J. Funct. Anal. 73(2): 369–429

    Article  MATH  MathSciNet  Google Scholar 

  3. Brüning J. and Seeley R. (1988). An index theorem for first order regular singular operators. Am. J. Math. 110(4): 659–714

    Article  MATH  Google Scholar 

  4. Brüning J. and Seeley R. (1991). The expansion of the resolvent near a singular stratum of conical type. J. Funct. Anal. 95(2): 255–290

    Article  MATH  MathSciNet  Google Scholar 

  5. Callias C. (1983). The heat equation with singular coefficients. I. Operators of the form −d 2/dx 2 +  κ/x 2 in dimension 1. Comm. Math. Phys. 88(3): 357–385

    Article  MATH  MathSciNet  Google Scholar 

  6. Callias C. and Uhlmann G. (1984). Singular asymptotics approach to partial differential equations with isolated singularities in the coefficients. Bull. Am. Math. Soc. (N.S.) 11(1): 172–176

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheeger J. (1979). On the spectral geometry of spaces with cone-like singularities. Proc. Natl. Acad. Sci. USA 76: 2103–2106

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheeger J. (1983). Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18(4): 575–657

    MATH  MathSciNet  Google Scholar 

  9. Chou A. (1985). The Dirac operator on spaces with conical singularities and positive scalar curvatures. Trans. Am. Math. Soc. 289(1): 1–40

    Article  MATH  Google Scholar 

  10. Falomir H., Muschietti M.A. and Pisani P.A.G. (2004). On the resolvent and spectral functions of a second order differential operator with a regular singularity. J. Math. Phys. 45(12): 4560–4577

    Article  MATH  MathSciNet  Google Scholar 

  11. Falomir H., Muschietti M.A., Pisani P.A.G. and Seeley R. (2003). Unusual poles of the ζ-functions for some regular singular differential operators. J. Phys. A 36(39): 9991–10010

    Article  MATH  MathSciNet  Google Scholar 

  12. Fedosov B., Schulze B.-W. and Tarkhanov N. (1999). The index of elliptic operators on manifolds with conical points. Selecta Math. (N.S.) 5(4): 467–506

    Article  MATH  MathSciNet  Google Scholar 

  13. Fox J. and Haskell P. (1995). Index theory for perturbed Dirac operators on manifolds with conical singularities. Proc. Am. Math. Soc. 123(7): 2265–2273

    Article  MATH  MathSciNet  Google Scholar 

  14. Gil J.B. (2003). Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators. Math. Nachr. 250: 25–57

    Article  MATH  MathSciNet  Google Scholar 

  15. Gil J. and Loya P. (2002). On the noncommutative residue and the heat trace expansion on conic manifolds. Manuscr. Math. 109(3): 309–327

    Article  MATH  MathSciNet  Google Scholar 

  16. Gil J. and Mendoza G. (2003). Adjoints of elliptic cone operators. Am. J. Math. 125(2): 357–408

    Article  MATH  MathSciNet  Google Scholar 

  17. Gil, J., Loya, P., Mendoza, G.: A note on the index of cone differential operators, preprint arXiv:math/0110172v1 [math.AP], 2001

  18. Gil J., Krainer T. and Mendoza G. (2006). Resolvents of elliptic cone operators. Funct. Anal. 241(1): 1–55

    Article  MATH  MathSciNet  Google Scholar 

  19. Grieser, D., Gruber, M.: Singular asymptotics lemma and push-forward theorem. In: Approaches to singular analysis (Berlin, 1999). Oper. Theory Adv. Appl., vol. 125, pp. 117–130. Birkhäuser, Basel (2001)

  20. Grubb G. and Seeley R. (1995). Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems. Invent. Math. 121: 481–529

    Article  MATH  MathSciNet  Google Scholar 

  21. Kozlov, V., Maz’ya, V., Roßmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations. In: Mathematical Surveys and Monographs, vol. 85. AMS, Providence (2001)

  22. Lesch, M.: Operators of Fuchs type, conical singularities, and asymptotic methods. In: Teubner-Texte zur Math., vol. 136. B.G. Teubner, Stuttgart, Leipzig (1997)

  23. Lescure J.-M. (2001). Triplets spectraux pour les variétés à singularité conique isolée. Bull. Soc. Math. France 129(4): 593–623

    MATH  MathSciNet  Google Scholar 

  24. Loya P. (2002). On the resolvent of differential operators on conic manifolds. Comm. Anal. Geom. 10(5): 877–934

    MATH  MathSciNet  Google Scholar 

  25. Loya P. (2002). Parameter-dependent operators and resolvent expansions on conic manifolds. Ill. J Math. 46(4): 1035–1059

    MATH  MathSciNet  Google Scholar 

  26. Loya P. (2003). Asymptotic properties of the heat kernel on conic manifolds. Isr. J. Math. 136: 285–306

    Article  MATH  MathSciNet  Google Scholar 

  27. Mazzeo R. (1991). Elliptic theory of differential edge operators. I. Comm. Partial Differ. Equ. 16(10): 1615–1664

    Article  MATH  MathSciNet  Google Scholar 

  28. Melrose R. (1992). Calculus of conormal distributions on manifolds with corners. Int. Math. Res. Not. 3: 51–61

    Article  MathSciNet  Google Scholar 

  29. Melrose, R.: The Atiyah-Patodi-Singer index theorem. In: Research Notes in Mathematics. A K Peters, Ltd., Wellesley (1993)

  30. Melrose R. (1995). The eta invariant and families of pseudodifferential operators. Math. Res. Lett. 2(5): 541–561

    MATH  MathSciNet  Google Scholar 

  31. Melrose, R., Mendoza, G.: Elliptic operators of totally characteristic type, MSRI preprint (1983)

  32. Melrose, R., Nistor, V.: Homology of pseudodifferential operators I. Manifolds with boundary, preprint (1996)

  33. Mooers E. (1999). Heat kernel asymptotics on manifolds with conic singularities. J. Anal. Math. 78: 1–36

    Article  MATH  MathSciNet  Google Scholar 

  34. Piazza P. (1993). On the index of elliptic operators on manifolds with boundary. J. Funct. Anal. 117(2): 308–359

    Article  MATH  MathSciNet  Google Scholar 

  35. Schrohe E. and Seiler J. (2005). The resolvent of closed extensions of cone differential operators. Can. J. Math. 57(4): 771–811

    MATH  MathSciNet  Google Scholar 

  36. Schulze, B.-W.: Pseudo-differential operators on manifolds with edges. In: Proceedings of the Symposium on ‘Partial Differential Equations’, Holzhau 1988 (Leipzig), Teubner-Texte zur Math., Teubner, vol. 112, pp. 259–288 (1989)

  37. Schulze B.-W. (1991). Pseudo-differential operators on manifolds with singularities. North-Holland, Amsterdam

    MATH  Google Scholar 

  38. Schulze B.-W., Sternin B. and Shatalov V. (1998). On the index of differential operators on manifolds with conical singularities. Ann. Glob. Anal. Geom. 16(2): 141–172

    Article  MATH  MathSciNet  Google Scholar 

  39. Seeley R. (1967). Complex powers of an elliptic operator. AMS Symp. Pure Math. 10: 288–307

    MathSciNet  Google Scholar 

  40. Seeley, R.: Topics in pseudo-differential operators. In: CIME Conference of Pseudo-Differential Operators 1968, Edizione Cremonese, Roma, pp. 169–305 (1969)

  41. Shubin M. (2001). Pseudodifferential operators and spectral theory, translated from the 1978 Russian original by Stig I. Andersson. Springer, Berlin

    Google Scholar 

  42. Witt, I.: On the factorization of meromorphic Mellin symbols, Parabolicity, Volterra calculus, and conical singularities. In: Oper. Theory Adv. Appl., vol. 138, pp. 279–306. Birkhäuser, Basel (2002)

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Authors and Affiliations

  1. Penn State Altoona, 3000 Ivyside Park, Altoona, PA, 16601-3760, USA

    Juan B. Gil

  2. Department of Mathematics, Binghamton University, Binghamton, NY, 13902, USA

    Paul A. Loya

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  1. Juan B. Gil
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  2. Paul A. Loya
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Correspondence to Juan B. Gil.

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Gil, J.B., Loya, P.A. Resolvents of cone pseudodifferential operators, asymptotic expansions and applications. Math. Z. 259, 65–95 (2008). https://doi.org/10.1007/s00209-007-0212-6

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  • DOI: https://doi.org/10.1007/s00209-007-0212-6

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