Heat traces and spectral zeta functions for $p$-adic Laplacians

L. F. Chacón-Cortés ; W. A. Zúñiga-Galindo

Algebra i analiz, Tome 29 (2017) no. 3, pp. 144-166.

Résumé

The study of the heat traces and spectral zeta functions for certain $p$-adic Laplacians is initiated. It is shown that the heat traces are given by $p$-adic integrals of Laplace type, and that the spectral zeta functions are $p$-adic integrals of Igusa type. Good estimates are found for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.

Détail

Keywords: heat traces, spectral zeta functions, Minakshisundaram–Pleijel zeta functions, $p$-adic heat equation, $p$-adic functional analysis.

Bibliographie

[1] Albeverio S., Karwowski W., Yasuda K., “Trace formula for $p$-adics”, Acta Appl. Math., 71:1 (2002), 31–48 | DOI | MR | Zbl

[2] Albeverio S., Khrennikov A. Yu., Shelkovich V. M., Theory of $p$-adic distributions: linear and nonlinear models, London Math. Soc. Lecture Note Ser., 370, Cambridge Univ. Press, Cambridge, 2010 | MR | Zbl

[3] Arendt W., Nittka R., Peter W., Steiner F., “Weyl's law: Spectral properties of the laplacian in mathematical physics”, Mathematical Analysis of Evolution, Information, and Complexity, Wiley-VCH, Weinheim, 2009, 1–71 | Zbl

[4] Avetisov V. A., Bikulov A. Kh., Osipov V. A., “$p$-adic description of characteristic relaxation in complex systems”, J. Phys. A, 36:15 (2003), 4239–4246 | DOI | MR | Zbl

[5] Avetisov V. A., Bikulov A. H., Kozyrev S. V., Osipov V. A., “$p$-adic models of ultrametric diffusion constrained by hierarchical energy landscapes”, J. Phys. A, 35:2 (2002), 177–189 | DOI | MR | Zbl

[6] Burnol J.-F., “Scattering on the $p$-adic field and a trace formula”, Int. Math. Res. Not. IMRN, 2000:2 (2000), 57–70 | DOI | MR | Zbl

[7] Casas-Sánchez O. F., Zúñiga-Galindo W. A., “$p$-adic elliptic quadratic forms, parabolic-type pseudodifferential equations with variable coefficients and Markov processes”, $p$-Adic Numbers Ultrametric Anal. Appl., 6:1 (2014), 1–20 | DOI | MR | Zbl

[8] Cazenave T., Haraux A., An introduction to semilinear evolution equations, Oxford Lecture Ser. Math. Appl., 13, Oxford Univ. Press, New York, 1998 | MR | Zbl

[9] Chacón-Cortés L. F., Zúñiga-Galindo W. A., “Nonlocal operators, parabolic-type equations, and ultrametric random walks”, J. Math. Phys., 54:11 (2013), 113503 ; “Erratum”, J. Math. Phys., 55:10 (2014), 109901 | DOI | MR | Zbl | DOI | MR | Zbl

[10] Chacón-Cortés L. F., Zúñiga-Galindo W. A., “Non-local operators, non-Archimedean parabolic-type equations with variable coefficients and Markov processes”, Publ. Res. Inst. Math. Sci., 51:2 (2015), 289–317 | DOI | MR | Zbl

[11] Chambert-Loir A., Tschinkel Yu., “Igusa integrals and volume asymptotics in analytic and adelic geometry”, Confluentes Math., 2:3 (2010), 351–429 | DOI | MR | Zbl

[12] Connes A., “Trace formula in non-commutative geometry and the zeros of the Riemann zeta function”, Selecta Math. (N.S.), 5 (1999), 29–106 | DOI | MR | Zbl

[13] Deninger Ch., “On the nature of the “explicit formulas” in analytic number theory – a simple example”, Number Theoretic Methods (Iizuka, 2001), Dev. Math., 8, Kluwer Acad. Publ., Dordrecht, 2002, 97–118 | MR | Zbl

[14] Dodziuk J., “Eigenvalues of the Laplacian and the heat equation”, Amer. Math. Monthly, 88:9 (1981), 686–695 | DOI | MR | Zbl

[15] Dragovich B., Khrennikov A. Yu., Kozyrev S. V., Volovich I. V., “On $p$-adic mathematical physics”, $p$-Adic Numbers Ultrametric Anal. Appl., 1:1 (2009), 1–17 | DOI | MR | Zbl

[16] Gilkey P. B., Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Stud. Adv. Math., Second ed., CRC Press, Boca Raton, FL, 1995 | MR | Zbl

[17] Haran Sh., “Quantizations and symbolic calculus over the $p$-adic numbers”, Ann. Inst. Fourier (Grenoble), 43:4 (1993), 997–1053 | DOI | MR | Zbl

[18] Igusa J.-I., An introduction to the theory of local zeta functions, AMS/IP Stud. Adv. Math., 14, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl

[19] Kochubei A. N., Pseudo-differential equations and stochastics over non-Archimedean fields, Monogr. and Textbook Pure Appl. Math., 244, Marcel Dekker, Inc., New York, 2001 | MR | Zbl

[20] Kozyrev S. V., “Teoriya vspleskov kak $p$-adicheskii spektralnyi analiz”, Izv. RAN. Ser. mat., 66:2 (2002), 149–158 | DOI | MR | Zbl

[21] Kozyrev S. V., Khrennikov A. Yu., “Psevdodifferentsialnye operatory na ultrametricheskikh prostranstvakh i ultrametricheskie vspleski”, Izv. RAN. Ser. mat., 69:5 (2005), 133–148 | DOI | MR | Zbl

[22] Lal N., Lapidus M. L., “Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals”, J. Phys. A, 45:36 (2012), 365205 | DOI | MR | Zbl

[23] Leichtnam E., “Scaling group flow and Lefschetz trace formula for laminated spaces with $p$-adic transversal”, Bull. Sci. Math., 131:7 (2007), 638–669 | DOI | MR | Zbl

[24] Leichtnam E., “On the analogy between arithmetic geometry and foliated spaces”, Rend. Mat. Appl. (7), 28:2 (2008), 163–188 | MR | Zbl

[25] Minakshisundaram S., “A generalization of Epstein zeta functions”, Canad. J. Math., 1 (1949), 320–327 | DOI | MR

[26] Minakshisundaram S., “Eigenfunctions on Riemannian manifolds”, J. Indian Math. Soc. (N.S.), 17 (1953), 159–165 | MR

[27] Minakshisundaram S., Pleijel E., “Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds”, Canad. J. Math., 1 (1949), 242–256 | DOI | MR | Zbl

[28] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional analysis, Acad. Press, New York–London, 1972 | MR | Zbl

[29] Riesz F., Sz.-Nagy B., Functional analysis, Dover Books Adv. Math., Dover Publ., Inc., New York, 1990 | MR | Zbl

[30] Rodríguez-Vega J. J., Zúñiga-Galindo W. A., “Taibleson operators, $p$-adic parabolic equations and ultrametric diffusion”, Pacific J. Math., 237:2 (2008), 327–347 | DOI | MR | Zbl

[31] Shubin M. A., Psevdodifferentsialnye operatory i spektralnaya teoriya, Izd-vo MGU, M., 1978 | MR

[32] Steinhurst B. A., Teplyaev A., “Existence of a meromorphic extension of spectral zeta functions on fractals”, Lett. Math. Phys., 103:12 (2013), 1377–1388 | DOI | MR | Zbl

[33] Taibleson M. H., Fourier analysis on local fields, Princeton Univ. Press, Princeton, 1975 | MR | Zbl

[34] Torba S. M., Zúñiga-Galindo W. A., “Parabolic type equations and Markov stochastic processes on adeles”, J. Fourier Anal. Appl., 19:4 (2013), 792–835 | DOI | MR | Zbl

[35] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR | Zbl

[36] Voros A., “Spectral zeta functions”, Zeta Functions in Geometry (Tokyo, 1990), Adv. Stud. Pure Math., 21, Kinokuniya, Tokyo, 1992, 327–358 | MR | Zbl

[37] Yasuda K., “Trace formula on the $p$-adic upper half-plane”, J. Funct. Anal., 216:2 (2004), 422–454 | DOI | MR | Zbl

[38] Zúñiga-Galindo W. A., “The non-Archimedean stochastic heat equation driven by Gaussian noise”, J. Fourier Anal. Appl., 21:3 (2015), 600–627 | DOI | MR | Zbl

[39] Zúñiga-Galindo W. A., “Parabolic equations and Markov processes over $p$-adic fields”, Potential Anal., 28:2 (2008), 185–200 | DOI | MR | Zbl