×

zbMATH — the first resource for mathematics

Pure point spectrum for the Laplacian on unbounded nested fractals. (English) Zbl 0965.35103
The author considers the Laplace operator on unbounded nested fractals which consist of self-similar and finitely ramified sets (invariant for a large group of symmetries) and shows that the set of Neumann-Dirichlet eigenvalues leads to pure-point spectrum with compactly supported eigenfunctions. The main results are given in three theorems which are proven by using only symmetry properties.

MSC:
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
28A80 Fractals
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Barlow, M.T.; Perkins, E.A., Brownian motion on the sierpinski gasket, Probab. theory related fields, 79, 543-623, (1988) · Zbl 0635.60090
[2] Barlow, M.T.; Kigami, J., Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London math. soc. (2), 56, 320-332, (1997) · Zbl 0904.35064
[3] Carmona, R.; Lacroix, J., Spectral theory of random Schrödinger operators, Probabilities and applications, (1990), Birkhaüser Boston
[4] Falconer, Fractal geometry: mathematical foundations and applications, (1990), Wiley Chichester · Zbl 0689.28003
[5] Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet forms and symmetric Markov processes, De gruyter studies in mathematics, 19, (1994), de Gruyter Berlin/New York
[6] Fukushima, M., Dirichlet forms, diffusion processes and spectral dimensions for nested fractals, (), 151-161 · Zbl 0764.60081
[7] Fukushima, M.; Shima, T., On the spectral analysis for the sierpinski gasket, Potential anal., 1, 1-35, (192) · Zbl 1081.31501
[8] Fukushima, M.; Shima, T., On the discontinuity and tail behaviours of the integrated density of states for nested pre-fractals, Comm. math. phys., 163, 461-471, (1994) · Zbl 0798.58049
[9] Fitzsimmons, P.J.; Hambly, B.M.; Kumagai, T., Transition density estimates for diffusions on affine nested fractals, Comm. math. phys., 165, 595-620, (1994) · Zbl 0853.60062
[10] Kigami, J., Harmonic calculus on p.c.f. self-similar sets, Trans. amer. math. soc., 335, 721-755, (1993) · Zbl 0773.31009
[11] Kigami, J., Distribution of localized eigenvalues of Laplacians on post-critically finite self-similar sets, J. funct. anal., 159, 170-198, (1998) · Zbl 0908.35087
[12] Kigami, J.; Lapidus, M.L., Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. math. phys., 158, 93-125, (1993) · Zbl 0806.35130
[13] S. Kusuoka, Lecture on Diffusion Processes on Nested Fractals, Lecture Notes in Mathematics, Vol, 1567, Springer-Verlag, New York/Berlin.
[14] Lapidus, M.L., Vibration of fractal drums, the Riemann hypothesis, waves in fractal media and the Weyl-Berry conjecture, Ordinary and partial differential equations, Research notes in maths, IV, (1993), Longman London, p. 126-209 · Zbl 0830.35094
[15] Lindstrøm, T., Brownian motion on nested fractals, Mem. amer. math. soc., 420, (1990) · Zbl 0688.60065
[16] Rammal, R., Spectrum of harmonic excitations on fractals, J. phys., 45, 191-206, (1984)
[17] Rammal, R.; Toulouse, G., J. phys. lett., 44, L-13, (1983)
[18] Sabot, C., Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. sci. ecole norm. sup. (4), 30, 605-673, (1997) · Zbl 0924.60064
[19] Sabot, C., Espaces de Dirichlet reliés par des points et application aux diffusions sur LES fractals finiment ramifiés, Potential anal., 11, 183-212, (1999) · Zbl 0945.60065
[20] Sabot, C., Density of states of diffusions on self-similar sets and holomorphic dynamics in P^k: the example of the interval [0, 1], C. R. acad. sci. Paris Sér. I math., 327, 359-364, (1998) · Zbl 0922.60040
[21] C. Sabot, Spectral analysis of a self-similar Sturm-Liouville operator, in preparation. · Zbl 1090.34070
[22] Strichartz, R.S., Fractals in the large, Canad. math. J., 50, 638-657, (1998) · Zbl 0913.28005
[23] J. P. Serre, Linear Representations of Finite Groups, Graduated Texts in Mathematics, Springer-Verlag, New York/Berlin. · Zbl 0223.20003
[24] Teplyaev, A., Spectral analysis on infinite sierpinski gasket, J. funct. anal., 159, 537-567, (1998) · Zbl 0924.58104
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.