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Zeta determinants on cones and products

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Abstract

We study the zeta determinant of the Laplace operator on two classes of (compact) geometries: cones and product spaces. The result is obtained by applying a new technique introduced in Spreafico (arXiv:math/0607816, 2008) to deal with some class of regularized products.

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References

  1. Bordag M., Geyer B., Kirsten K., Elizalde E.: Zeta function determinant of the Laplace operator on the D-dimensional ball. Commun. Math. Phys. 179, 215–234 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bordag M., Elizalde E., Kirsten K.: Heat kernel coefficients of the Laplace operator on the D-dimensional ball. J. Math. Phys. 37, 895–916 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bordag M., Kirsten K., Dowker S.: Heat kernels and functional determinants on a generalized cone. Commun. Math. Phys. 182, 371–394 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Cheeger J.: Spectral geometry of singular Riemannian spaces. J. Differ. Geom. 18, 575–657 (1984)

    MathSciNet  Google Scholar 

  6. Cheeger J.: Hodge theory of complex cones. Asterisque 101–102, 118–134 (1983)

    MathSciNet  Google Scholar 

  7. Cognola G., Zerbini S.: Zeta function on a generalized cone. Lett. Math. Phys. 42, 95–101 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dowker S., Kirsten K.: Spinors and forms on the ball and the generalized cone. Commun. Anal. Geom. 7, 641–679 (1999)

    MATH  MathSciNet  Google Scholar 

  9. Elizalde E., Odintsov S.D., Romeo A., Bytsenko A.A., Zerbini S.: Zeta Regularization Techniques with Applications. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  10. Elizalde E., Bordag M., Kirsten K.: Casimir energy for a massive fermionic quantum field with a spherical boundary. J. Phys. A 31, 1743–1759 (1998)

    Article  MATH  Google Scholar 

  11. Elizalde E.: Multidimensional extension of the generalized Chowla–Selberg formula. Commun. Math. Phys. 198, 83–95 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Elizalde E.: Ten physical applications of spectral zeta functions. In: Lecture Notes in Physics. Springer, Berlin (1995)

  13. Furutani K., de Gosson S.: Determinants of Laplacians on Heisenberg manifolds. J. Geom. Phys. 48, 438–479 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gilkey, P.B.: Invariance theorems, the heat equation, and the Atiyah–Singer index theorem. In: Studies in Advanced Mathematics. CRC Press, Boca Raton (1995)

  15. Jorgenson J., Lang S.: Artin formalism and heat kernels. J. Reine Angew. Math. 447, 165–200 (1994)

    MATH  MathSciNet  Google Scholar 

  16. Olver F.W.J.: Asymptotics and Special Functions. AKP Classics, New York (1997)

    MATH  Google Scholar 

  17. Ortenzi G., Spreafico M.: Zeta function regularization for a scalar field in a compact domain. J. Phys. A Math. Gen. 37, 11499–11517 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Spreafico M.: Zeta function and regularized determinant on projective spaces. Rocky Mount. J. Math. 33, 1499–1512 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Spreafico M.: Zeta function and regularized determinant on a disc and on a cone. J. Geom. Phys. 54, 355–371 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Spreafico M.: Zeta invariants for sequences of spectral type, special functions and the Lerch formula. Proc. R. Soc. Edinburgh 136A, 863–887 (2006)

    Article  MathSciNet  Google Scholar 

  21. Spreafico M.: Zeta invariants for Dirichlet series. Pac. J. Math. 224, 180–199 (2006)

    Article  MathSciNet  Google Scholar 

  22. Spreafico, M.: Zeta invariants for double sequences of spectral type (2008, preprint). arXiv:math/0607816

  23. Spreafico M., Zerbini S.: Spectral analysis and zeta determinants on the deformed spheres. Commun. Math. Phys 273, 677–704 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Mauro Spreafico.

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Spreafico, M. Zeta determinants on cones and products. J. Geom. 96, 167–178 (2009). https://doi.org/10.1007/s00022-010-0034-2

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  • DOI: https://doi.org/10.1007/s00022-010-0034-2

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