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Complex powers of elliptic pseudodifferential operators

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Abstract

The aim of this paper is the construction of complex powers of elliptic pseudodifferential operators and the study of the analytic properties of the corresponding kernels kS (x,y). For x=y, the case of principal interest, the domain of holomorphy and the singularities of kS (x,x) are shown to depend on the asymptotic expansion of the symbol. For classical symbols, kS (x,x) is known to be meromorphic on ℂ with simple poles in a set of equidistant points on the real axis. In the more general cases considered here, the singularities may be distributed over a half plane and kS (x,x) can not always be extended to337-2. An example is given where kS (x,x) has a vertical line as natural boundary.

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References

  1. DOETSCH, G.: Handbuch der Laplace-Transformation, Birkhäuser, Basel 1950

    Google Scholar 

  2. GILKEY, P.: The Residue of the Global η Function at the Origin, Adv. Math. 40. 290–307 (1981)

    Google Scholar 

  3. GILKEY, P.: The Index Theorem and the Heat Equation, Publish or Perish, Boston 1974

    Google Scholar 

  4. KUMANO-GO, H.: Pseudodifferential Operators, MIT Press, Cambridge and London 1981

    Google Scholar 

  5. SEELEY, R.T.: Complex Powers of an Elliptic Operator, AMS Proc. Symp. Pure Math. X, 288–307

  6. SEELEY, R. T.: The Resolvent of an Elliptic Boundary Problem, A.J.M. 91, 889–920 (1969)

    Google Scholar 

  7. SEELEY, R.T.: Topics in Pseudo-differential Operators, CIME IIo ciclo 1968, Ed. Cremonese, Rome 1969

    Google Scholar 

  8. SUBIN, M.A.: Pseudodifferential Operators and Spectral Theory, Nauka, Moscow 1978 (russian)

    Google Scholar 

  9. WODZICKI, M.: Spectral asymmetry and Zeta Functions, Inv. Math. 66, 115–135 (1982)

    Google Scholar 

  10. WIDOM, H.: A Complete Symbolic Calculus for Pseudo-differential Operators, Bull Sc. Math. 104 (1980), 19–64

    Google Scholar 

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Authors and Affiliations

  1. Fachbereich Mathematik, Johannes Gutenberg-Universität, 6500, Mainz, Germany

    Elmar Schrohe

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  1. Elmar Schrohe
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Dedicated to Professor H.G. Tillmann on the occasion of his 60th birthday

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Schrohe, E. Complex powers of elliptic pseudodifferential operators. Integr equ oper theory 9, 337–354 (1986). https://doi.org/10.1007/BF01199350

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  • DOI: https://doi.org/10.1007/BF01199350

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