Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

Okoudjou, Kasso; Strichartz, Robert

doi:10.1090/S0002-9939-07-09008-9

Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket
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by Kasso A. Okoudjou and Robert S. Strichartz PDF

Proc. Amer. Math. Soc. 135 (2007), 2453-2459 Request permission

Abstract:

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpiński gasket $SG$. In particular, using the existence of localized eigenfunctions for the Laplacian on $SG$ we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

References

Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995) Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1–121. MR 1668115, DOI 10.1007/BFb0092537
Martin T. Barlow and Jun Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc. (2) 56 (1997), no. 2, 320–332. MR 1489140, DOI 10.1112/S0024610797005358
Kevin Coletta, Kealey Dias, and Robert S. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs’ phenomenon, Fractals 12 (2004), no. 4, 413–449. MR 2109985, DOI 10.1142/S0218348X04002689
Kyallee Dalrymple, Robert S. Strichartz, and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5 (1999), no. 2-3, 203–284. MR 1683211, DOI 10.1007/BF01261610
M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal. 1 (1992), no. 1, 1–35. MR 1245223, DOI 10.1007/BF00249784
V. Guillemin, Some spectral results on rank one symmetric spaces, Advances in Math. 28 (1978), no. 2, 129–137. MR 494331, DOI 10.1016/0001-8708(78)90059-2
V. Guillemin, Some spectral results on rank one symmetric spaces, Advances in Math. 28 (1978), no. 2, 129–137. MR 494331, DOI 10.1016/0001-8708(78)90059-2
Victor Guillemin, Some spectral results for the Laplace operator with potential on the $n$-sphere, Advances in Math. 27 (1978), no. 3, 273–286. MR 478245, DOI 10.1016/0001-8708(78)90102-0
Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), no. 2, 721–755. MR 1076617, DOI 10.1090/S0002-9947-1993-1076617-1
Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042, DOI 10.1017/CBO9780511470943
R. Rammal and G. Toulouse, Random walks on fractal structures and percoloration clusters, J. Physique Lett., 44 (1983), L13–L22.
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Robert S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46 (1999), no. 10, 1199–1208. MR 1715511
Robert S. Strichartz, Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006. A tutorial. MR 2246975
Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal. 159 (1998), no. 2, 537–567. MR 1658094, DOI 10.1006/jfan.1998.3297
Alan Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), no. 4, 883–892. MR 482878