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Asymptotics of the determinant of the discrete Laplacian on the circle

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Abstract

We compute the coefficients of the terms with non negative powers of the asymptotic expansion of the determinant of the discrete Laplace operator of the de Rham complex of the circle associated to any orthogonal representation of the fundamental group.

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References

  1. Chaumard, L.: Discretisation de zeta-determinants d’operateurs de Schrödinger sur le tore. Bull. Soc. Math. France 134, 327–355 (2006)

  2. Chinta, G., Jorgenson, J., Karlsson, A.: Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori. Nagoya Math. J. 198, 121–172 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dodziuk, J.: Finite-difference approach to the Hodge theory of harmonic forms. Am. J. Math. 98, 79–104 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dodziuk, J., Patodi, V.K.: Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. 40, 1–52 (1976)

    MathSciNet  MATH  Google Scholar 

  5. Eckmann, B.: Harmonische Funktionen und Randvertanfgaben in einem Komplex. Commentarii Math. Helvetici 17, 240–245 (1944-45)

  6. Müller, W.: Analytic torsion and R-torsion of Riemannian manifolds. Adv. Math. 28, 233–305 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ray, D.B., Singer, I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  8. Seeley, R.: Complex powers of an elliptic operator, Singular integrals. In: Proceedings of Symposia in Pure Mathematics, Chicago, 1966, pp. 288–307. American Mathematical Society, Providence, R.I. (1967)

  9. Szegö, G.: Ein Grenzwertsatz über die Toeplitzschen Determinanten einer reellen positiven Funktion. Math. Ann. 76, 490–503 (1915)

    Article  MathSciNet  MATH  Google Scholar 

  10. Vertmann, B.: Regularized limit of determinants for discrete tori. arXiv: 1502.04541v1 (2015)

  11. Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis. Cambridge University Press, Cambridge (1946)

    MATH  Google Scholar 

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Acknowledgements

The author thanks the referee for his comments and remarks.

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Authors and Affiliations

  1. Dipartimento di matematica e fisica E. De Giorgi, Università del Salento, Lecce, Italy

    M. Spreafico

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  1. M. Spreafico
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Correspondence to M. Spreafico.

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Spreafico, M. Asymptotics of the determinant of the discrete Laplacian on the circle. Rev Mat Complut 31, 237–245 (2018). https://doi.org/10.1007/s13163-017-0233-6

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  • DOI: https://doi.org/10.1007/s13163-017-0233-6

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