Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

Proceedings of the London Mathematical Society

Volumn 72, Issue 1, 1996, Pages 188-214

  Levitin, Michael ^a   Vassiliev, Dmitri ^b  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF SUSSEX   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33747018374     PISSN: 00246115     EISSN: None     Source Type: Journal    
DOI: 10.1112/plms/s3-72.1.188     Document Type: Article

References (53)

1. M. VAN DEN BERG, 'Heat content and Brownian motion for some regions with a fractal boundary', Probab. Theory Related Fields 100 (1994) 439-456.
2. M. VAN DEN BERG, 'Heat content asymptotics for some open sets with a fractal boundary', Proceedings of the Summer Research Institute, Cornell, Ithaca, 1993, Proceedings of Symposia in Pure Mathematics 57 (American Mathematical Society, Providence, R.I., 1995), pp. 11-22.
3. m', Math. Z. 202 (1989) 463-482.
4. M. VAN DEN BERG and P. B. GILKEY, 'Heat content asymptotics of a Riemannian manifold with boundary', J. Funct. Anal. 120 (1994) 48-71.
5. M. VAN DEN BERG and J.-F. LE GALL, 'Mean curvature and the heat equation', Math. Z. 215 (1994) 437-464.
6. M. VAN DEN BERG and M. LEVITIN, 'Sharp asymptotics for functions of Weierstrass type and spectral asymptotics for iterated sets with fractal boundary', preprint, Heriot-Watt University, Edinburgh, 1994.
7. 2 with a polygonal boundary', J. London Math. Soc. (2) 37 (1988) 119-127.
8. 2 with a polygonal boundary', Probab. Theory Related Fields 86 (1990) 41-52.
9. M. V. BERRY, 'Distributions of modes in fractal resonators', Structural stability in physics (eds W. Güttinger and H. Eikeimeier, Springer, Berlin, 1979), pp. 51-53.
10. M. V. BERRY, 'Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals', Geometry of the Laplace operator, Proceedings of Symposia in Pure Mathematics 36 (American Mathematical Society, Providence, R.I., 1980), pp. 13-38.
11. M. S. BIRMAN and M. Z. SOLOMYAK, 'On the principal term of the spectral asymptotics for non-smooth elliptic problems', Functional Anal. Appl. 4 (1970).
12. J. BROSSARD and R. CARMONA, 'Can one hear the dimension of a fractal?', Comm. Math. Phys. 104 (1986) 103-122.
13. A. M. CAETANO, 'Some domains where the eigenvalues of the Dirichlet Laplacian have non-power second term asymptotic estimates', J. London Math. Soc. (2) 43 (1991) 431-450.
14. A. M. CAETANO, 'On the search for the asymptotic behaviour of the eigenvalues of the Dirichlet Laplacian for bounded irregular domains', Partial differential equations and fractals, Proceedings of the International Workshop in Toulouse, March 1993 (to appear 1995).
15. E. B. DAVIS, Heat kernels and spectral theory (Cambridge University Press, 1989).
16. K. J. FALCONER, Fractal geometry, mathematical foundations and applications (John Wiley, Chichester, 1990).
17. K. J. FALCONER, 'On the Minkowski measurability of fractals', Proc. Amer. Math. Soc. to appear.
18. W. FELLER, An introduction to probability theory and its applications, vol. 2, 2nd revised edn (John Wiley, New York, 1966).
19. J. FLECKINGER and M. L. LAPIDUS, '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) 171-175.
20. J. FLECKINGER, M. LEVITIN, and D. VASSILIEV, 'Heat content of the triadic von Koch snowflake', Partial differential equations and fractals, Proceedings of the International Workshop in Toulouse, March 1993 (to appear 1995).
21. J. FLECKINGER, M. LEVITIN, and D. VASSILIEV, 'Heat equation on the triadic von Koch snowflake: asymptotic and numerical analysis', Proc. London Math. Soc. (3) 71 (1995) 372-396.
22. J. FLECKINGER and G. MÉTIVIER, 'Théorie spectrale des opérateurs uniformément elliptiques sur quelques ouverts irréguliers', C. R. Acad. Sci. Paris Sér. A 276 (1973) 913-916.
23. J. FLECKINGER and D. VASSILIEV, 'Tambour fractal: exemple d'une formule asymptotique à deux termes pour la "fonction des comptage"', C. R. Acad. Sci. Paris Sér. I Math. 311 (1990) 867-872.
24. J. FLECKINGER-PELLE and D. VASSILIEV, 'An example of a two-term asymptotics for the "counting function" of a fractal drum', Trans. Amer. Math. Soc. 337 (1993) 99-116.
25. T.-Y. HU and K.-S. LAU, 'Fractal dimensions and singularities of the Weierstrass type functions', Trans. Amer. Math. Soc. 335 (1993) 649-665.
26. V. JA. IVRII, 'Second term of the spectral asymptotic expansion of the Laplace-Beltrami operator on manifolds with boundary', Functional. Anal. Appl. 14 (1980) 98-106.
27. V. JA. IVRII, Precise spectral asymptotics for elliptic operators acting in fiberings over manifolds with boundary, Lecture Notes in Mathematics 1100 (Springer, Berlin, 1984).
28. M. KAC, 'Can one hear the shape of a drum?', Amer. Math. Monthly 73 (1966) 1-23.
29. J. KIGAMI and M. L. LAPIDUS, 'Weyl's problem for the spectral distribution of the Laplacians on p.c.f. self-similar fractals', Comm. Math. Phys. 158 (1993) 93-125.
30. S. P. LALLEY, 'Renewal theorems in symbolic dynamics, with applications to geodesic flow, non-euclidian tesselations and their fractal limits', Acta Math. 163 (1989) 1-55.
31. M. 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) 465-529.
32. M. 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, Proceedings of the UAB International Conference on Mathematical Physics and Differential Equations, Birmingham, 1990 (eds C. Bennewitz et al., Academic Press, New York, 1992), pp. 151-182.
33. K.-S. LAU, 'Fractal measures and mean p-variations', J. Funct. Anal. 108 (1992) 427-457.
34. K.-S. LAU and J. WANG, 'Mean quadratic variations and Fourier asymptotics of self-similar measures', Monatsh. Math. 115 (1993) 99-132.
35. R. B. MELROSE, 'Weyl's conjecture for manifolds with concave boundary', Geometry of the Laplace Operator, Proceedings of Symposia in Pure Mathematics 36 (American Mathematical Society, Providence, R.I., 1980), pp. 257-273.
36. G. MÉTIVIER, 'Étude asymptotique des valeurs propres et de la fonction spectrale de problèmes aux limites', Thèse de doctorat d'Etat, Mathématiques, Université de Nice, France, 1976.
37. G. MÉTIVIER, 'Valeurs propres de problèmes aux limites elliptiques irréguliers', Bull. Soc. Math. France Mém. 51-52 (1977) 125-219.
38. d, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 837-842.
39. PHAM THE LAI, 'Meilleurs estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien', Math. Scand. 48 (1981) 5-38.
40. YU. G. SAFAROV, 'Precise asymptotics of the spectrum of a boundary value problem, and periodic billiards', Math. USSR-Izv. 52 (1988) 553-573.
41. 3', Adv. in Math. 29 (1978) 244-269.
42. R. T. SEELEY, 'An estimate near the boundary for the spectral function of the Laplace operator', Amer. J. Math. 102 (1980) 869-902.
43. B. SIMON, Functional integration and quantum physics (Academic Press, New York, 1979).
44. M. SOLOMYAK and E. VERBITSKY, 'On a spectral problem related to self-similar measures', Bull. London Math. Soc. 27 (1995) 242-248.
45. R. S. STRICHARTZ, 'Self-similar measures and their Fourier transforms', I, Indiana Univ. Math. J. 39 (1990) 797-817;
46. II, Trans. Amer. Math. Soc. 336 (1993) 335-361;
47. III, Indiana Univ. Math. J. 42 (1993) 367-411.
48. R. S. STRICHARTZ, 'Self-similarity in harmonic analysis', preprint, Cornell University, 1994.
49. D. VASSILIEV (VASIL'EV), 'Asymptotics of the spectrum of a boundary value problem', Trans. Moscow Math. Soc. 49 (1987) 173-245.
50. 2', Integral equations and inverse problems (eds V. Petkov and A. Lazarov, Longman, London, 1991), pp. 270-273.
51. H. WEYL, 'Über die asymptotische Verteilung der Eigenwerte', Gött. Nach. (1911) 110-117.
52. H. WEYL, 'Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen', Math. Ann. 71 (1912) 441-479.
53. M. WOODROOFE, Nonlinear renewal theory in sequential analysis (Society for Industrial and Applied Mathematics, Philadelphia, 1982).