Skip to main content
SpringerLink
Log in

Complex powers of differential operators on manifolds with conical singularities

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

We construct the complex powersA z for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatA z exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofA z , both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(A z ).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Bleistein and R. A. Handelsman,Asymptotic Expansions of Integrals, Dover, New York, 1986.

    Google Scholar 

  2. B. Bucicovschi,An extension of the work of V. Guillemin on complex powers and zeta functions of elliptic pseudodifferential operators, Proc. Amer. Math. Soc.127 (1999), 3081–3090.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Coriasco, E. Schrohe and J. Seiler,Bounded imaginary powers of differential operators on manifolds with conical singularities, Math. Z., to appear.

  4. J. B. Gil,Full asymptotic expansion of the heat trace for non-self-adjoint elliptic cone operators, Math. Nachr., to appear.

  5. G. Grubb,Functional Calculus of Pseudodifferential Boundary Problems, 2nd edn., Birkhäuser, Boston, 1996.

    MATH  Google Scholar 

  6. V. Guillemin,A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. in Math.55 (1985), 131–160.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Kumano-go,Pseudodifferential Operators, MIT Press, Cambridge, Mass., 1981.

    Google Scholar 

  8. R. Lauter,On Ψ * and C *-algebras of pseudodifferential operators on manifolds with conical singularities, inBanach Algebras '97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 303–324.

    Google Scholar 

  9. R. Lauter and J. Seiler,Pseudodifferential analysis on manifolds with boundary—a comparison of b-calculus and cone algebra, inApproaches to Singular Analysis (Berlin, 1999), Birkhäuser, Basel, 2001, pp. 131–166.

    Google Scholar 

  10. P. Loya,Tempered operators and the heat kernel and complex powers of elliptic pseudodifferential operators, Comm. Partial Differential Equtions26 (2001), 1253–1321.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Loya,On the resolvent of differential operators on conic manifolds, Comm. Anal. Geom., to appear.

  12. P. Loya,Asymptotic properties of the heat kernel on conic manifolds, Israel J. Math., to appear.

  13. R. Mazzeo,Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations16 (1991), 1615–1664.

    Article  MATH  MathSciNet  Google Scholar 

  14. R. B. Melrose,The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, Wellesley, 1993.

    MATH  Google Scholar 

  15. E. Mooers,Heat kernel asymptotics on manifolds with conic singularities, J. Analyse Math.78 (1999), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Rempel and B.-W. Schulze,Complex powers for pseudodifferential boundary problems. I, Math. Nachr.111 (1983), 41–109.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Schrohe,Complex powers of elliptic pseudodifferential operators, Integral Equations Operator Theory9 (1986), 337–354.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Schrohe,Complex powers on noncompact manifolds and manifolds with singularities, Math. Ann.281 (1988), 393–409.

    Article  MATH  MathSciNet  Google Scholar 

  20. B.-W. Schulze,Boundary Value Problems and Singular Pseudodifferential Operators, Wiley, Chichester, 1998.

    Google Scholar 

  21. R. T. Seeley,Complex powers of an elliptic operator, Proc. Sympos. Pure Math.10 (1967). 288–307.

    MathSciNet  Google Scholar 

  22. M. A. Shubin,Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987.

    MATH  Google Scholar 

  23. H. Widom,A complete symbolic calculus for pseudodifferential operators, Bull. Sci. Math. (2)104 (1980), 19–63.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Binghamton University, PO Box 6000, 13902-6000, Binghamton, NY, USA

    Paul Loya

Authors
  1. Paul Loya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paul Loya.

Additional information

Supported by a Ford Foundation Fellowship administered by the National Research Council.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Loya, P. Complex powers of differential operators on manifolds with conical singularities. J. Anal. Math. 89, 31–56 (2003). https://doi.org/10.1007/BF02893076

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02893076

Keywords

Search

Navigation