Advertisement

Analytic Properties of Spectral Functions

  • Michał EcksteinEmail author
  • Bruno Iochum
Chapter
Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 27)

Abstract

In the previous chapter we have witnessed the interplay between the spectral zeta functions and the associated heat traces (cf. Proposition  2.10). We have also learned, in Sect.  2.2.2, how to exploit the Laplace transform to compute the spectral action from a given heat trace. In this chapter we further explore the connections between the spectral functions unravelling the intimate relationship between the meromorphic continuation of a zeta function and the asymptotic expansion of the corresponding heat trace. We utilise the latter to establish the sought asymptotic expansion of the spectral action at large energies. Finally, we ponder the possibility of obtaining convergent, rather than only asymptotic, formulae for this action.

References

  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Courier Dover Publications, USA (2012)zbMATHGoogle Scholar
  2. 2.
    Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976)zbMATHGoogle Scholar
  3. 3.
    Boeijink, J., van den Dungen, K.: On globally non-trivial almost-commutative manifolds. J. Math. Phys. 55(10), 103508 (2014)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. J. Am. Math. Soc. 30(1), 205–224 (2017).  https://doi.org/10.1090/jams/860MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ćaćić, B.: A reconstruction theorem for almost-commutative spectral triples. Lett. Math. Phys. 100(2), 181–202 (2012)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Carlson, B.C.: Special Functions of Applied Mathematics. Academic Press, Cambridge (1977)zbMATHGoogle Scholar
  7. 7.
    Chamseddine, A.H., Connes, A.: The uncanny precision of the spectral action. Commun. Math. Phys. 293(3), 867–897 (2009)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Connes, A.: Cyclic cohomology, quantum group symmetries and the local index formula for \(SU_q(2)\). J. Inst. Math. Jussieu 3(1), 17–68 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Connes, A., Marcolli, M.: Noncommutative Geometry. Quantum Fields and Motives. American Mathematical Society. Colloquium Publications, Providence (2008)zbMATHGoogle Scholar
  10. 10.
    Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. GAFA 5(2), 174–243 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Eckstein, M., Iochum, B., Sitarz, A.: Heat trace and spectral action on the standard Podleś sphere. Commun. Math. Phys. 332(2), 627–668 (2014)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Eckstein, M., Zając, A.: Asymptotic and exact expansions of heat traces. Math. Phys. Anal. Geom. 18(1), 1–44 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Elizalde, E., Leseduarte, S., Zerbini, S.: Mellin transform techniques for zeta-function resummations. arXiv:hep-th/9303126 (1993)
  14. 14.
    Flajolet, P., Sedgewick, R.: The average case analysis of algorithms: Mellin transform asymptotics (1996). Research report RR-2956, INRIAGoogle Scholar
  15. 15.
    Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Studies in Advanced Mathematics, 2nd edn. CRC Press, Boca Raton (1995)Google Scholar
  16. 16.
    Grubb, G., Seeley, R.T.: Weakly parametric pseudodifferential operators and Atiyah-Patodi-Singer boundary problems. Invent. Math. 121(1), 481–529 (1995)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Guido, D., Isola, T.: New results on old spectral triples for fractals. In: Alpay, D., Cipriani, F., Colombo, F., Guido, D., Sabadini, I., Sauvageot, J.L. (eds.) Noncommutative Analysis, Operator Theory and Applications, pp. 261–270. Birkhäuser, Basel (2016)CrossRefGoogle Scholar
  18. 18.
    Iochum, B., Levy, C., Sitarz, A.: Spectral action on \(SU_q(2)\). Commun. Math. Phys. 289(1), 107–155 (2009)ADSCrossRefGoogle Scholar
  19. 19.
    Lapidus, M.L., van Frankenhuijsen, M.: Fractal Geometry. Complex Dimensions and Zeta Functions. Springer, Berlin (2006)zbMATHGoogle Scholar
  20. 20.
    Lesch, M.: On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols. Ann. Glob. Anal. Geom. 17(2), 151–187 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Matsumoto, K., Weng, L.: Zeta-functions defined by two polynomials. In: Kanemitsu, S., Jia, C. (eds.) Number Theoretic Methods, pp. 233–262. Springer, Berlin (2002)CrossRefGoogle Scholar
  22. 22.
    Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals. Encyclopedia of mathematics and its applications, vol. 85. Cambridge University Press, Cambridge (2001)Google Scholar
  23. 23.
    Romano, J.P., Siegel, A.F.: Counterexamples in Probability and Statistics. Wadsworth & Brooks/Cole Advanced Books & Software (1986)Google Scholar
  24. 24.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  25. 25.
    Titchmarsh, E.C., Heath-Brown, D.R.: The Theory of the Riemann Zeta-function. Oxford University Press, Oxford (1986)Google Scholar
  26. 26.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1927)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Copernicus Center for Interdisciplinary StudiesKrakówPoland
  2. 2.Faculty of Physics, Astronomy and Applied Computer ScienceJagiellonian UniversityKrakówPoland
  3. 3.National Quantum Information Centre, Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and InformaticsUniversity of GdańskGdańskPoland
  4. 4.Aix-Marseille Univ, Université de Toulon, CNRS, CPTMarseilleFrance

Personalised recommendations