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An example of a two-term asymptotics for the “counting function” of a fractal drum


Authors: Jacqueline Fleckinger-Pellé and Dmitri G. Vassiliev
Journal: Trans. Amer. Math. Soc. 337 (1993), 99-116
MSC: Primary 58G18; Secondary 58G25
DOI: https://doi.org/10.1090/S0002-9947-1993-1176086-7
MathSciNet review: 1176086
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Abstract: In this paper we study the spectrum of the Dirichlet Laplacian in a bounded domain $\Omega \subset {\mathbb {R}^n}$ with fractal boundary $\partial \Omega$. We construct an open set $\mathcal {Q}$ for which we can effectively compute the second term of the asymptotics of the "counting function" $N(\lambda ,\mathcal {Q})$, the number of eigenvalues less than $\lambda$. In this example, contrary to the M. V. Berry conjecture, the second asymptotic term is proportional to a periodic function of In $\lambda$, not to a constant. We also establish some properties of the $\zeta$-function of this problem. We obtain asymptotic inequalities for more general domains and in particular for a connected open set $\mathcal {O}$ derived from $\mathcal {Q}$. Analogous periodic functions still appear in our inequalities. These results have been announced in $[{\text {FV}}]$.


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  • M. V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 13–28. MR 573427
  • M. Š. Birman and M. Z. Solomjak, The principal term of the spectral asymptotics for “non-smooth” elliptic problems, Funkcional. Anal. i Priložen. 4 (1970), no. 4, 1–13 (Russian). MR 0278126
  • Jean Brossard and René Carmona, Can one hear the dimension of a fractal?, Comm. Math. Phys. 104 (1986), no. 1, 103–122. MR 834484
  • R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, Inc., New York, N.Y., 1953. MR 0065391
  • J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI https://doi.org/10.1007/BF01405172
  • K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
  • Jacqueline Fleckinger and Guy Métivier, Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A913–A916 (French). MR 320550
  • Jacqueline Fleckinger and Dmitri G. Vasil′ev, Tambour fractal: exemple d’une formule asymptotique à deux termes pour la “fonction de comptage”, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 13, 867–872 (French, with English summary). MR 1084044
  • C. F. Gauss, Disquisitiones arithmeticae, Leipzig, 1801.
  • V. Ja. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 25–34 (Russian). MR 575202
  • Victor Ivriĭ, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Mathematics, vol. 1100, Springer-Verlag, Berlin, 1984. MR 771297
  • Michel L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), no. 2, 465–529. MR 994168, DOI https://doi.org/10.1090/S0002-9947-1991-0994168-5
  • Michel L. Lapidus, Spectral and fractal geometry: from the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function, Differential equations and mathematical physics (Birmingham, AL, 1990) Math. Sci. Engrg., vol. 186, Academic Press, Boston, MA, 1992, pp. 151–181. MR 1126694, DOI https://doi.org/10.1016/S0076-5392%2808%2963379-2
  • Michel L. Lapidus and Jacqueline Fleckinger, The vibrations of a “fractal drum”, Differential equations (Xanthi, 1987) Lecture Notes in Pure and Appl. Math., vol. 118, Dekker, New York, 1989, pp. 423–436. MR 1021743
  • Michel L. Lapidus and Jacqueline Fleckinger-Pellé, Tambour fractal: vers une résolution de la conjecture de Weyl-Berry pour les valeurs propres du laplacien, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 4, 171–175 (French, with English summary). MR 930556
  • Michel L. Lapidus and Carl Pomerance, Fonction zêta de Riemann et conjecture de Weyl-Berry pour les tambours fractals, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 6, 343–348 (French, with English summary). MR 1046509
  • R. B. Melrose, Weyl’s conjecture for manifolds with concave boundary, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 257–274. MR 573438
  • Richard Melrose, The trace of the wave group, Microlocal analysis (Boulder, Colo., 1983) Contemp. Math., vol. 27, Amer. Math. Soc., Providence, RI, 1984, pp. 127–167. MR 741046, DOI https://doi.org/10.1090/conm/027/741046
  • G. Métivier, Etude asymptotique des valeurs propres et de la fonction spectrale de problèmes aux limites, Thèse de Doctorat d’Etat, Mathématiques, Université de Nice, France, 1976.
  • Guy Métivier, Valeurs propres de problèmes aux limites elliptiques irrégulières, Bull. Soc. Math. France Suppl. Mém. 51-52 (1977), 125–219 (French). MR 473578
  • Yu. G. Safarov, Asymptotics of the spectrum of a boundary value problem with periodic billiard trajectories, Funktsional. Anal. i Prilozhen. 21 (1987), no. 4, 88–90 (Russian). MR 925085
  • ---, Precise asymptotics of the spectrum of a boundary value problem and periodic billiards, Izv. Akad. Nauk SSSR Math. Ser. 52 (1988), no. 6, 1230-1251; English transl, in Math. USSR-Izv.
  • D. G. Vasil′ev, Asymptotic behavior of the spectrum of a boundary value problem, Trudy Moskov. Mat. Obshch. 49 (1986), 167–237, 240 (Russian). MR 853539
  • ---, One can hear the dimension of a connected fractal in ${\mathbb {R}^2}$, Petkov & Lazarov, Integral Equations and Inverse Problems, Longman Academic, Scientific & Technical, 1991, pp. 270-273. H. Weyl, Über die asymptotische Verteilung der Eigenwerte, Gött. Nach. (1911), 110-117.
  • Hermann Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), no. 4, 441–479 (German). MR 1511670, DOI https://doi.org/10.1007/BF01456804

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