ArticlesNo Access

HOMOTOPIES OF EIGENFUNCTIONS AND THE SPECTRUM OF THE LAPLACIAN ON THE SIERPINSKI CARPET

Department of Mathematics, Cornell University, Ithaca, NY 14850-4201, USA

Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA

and

Department of Mathematics, Cornell University, Ithaca, NY 14850-4201, USA

Mailing address for delivery of offprints: Dr Robert S. Strichartz, Mathematics Department, Cornell University, Malott Hall, Ithaca, NY 14853, USA.

Search for more papers by this author

https://doi.org/10.1142/S0218348X10004750Cited by:5 (Source: Crossref)

Abstract

Consider a family of bounded domains Ω_t in the plane (or more generally any Euclidean space) that depend analytically on the parameter t, and consider the ordinary Neumann Laplacian Δ_t on each of them. Then we can organize all the eigenfunctions into continuous families with eigenvalues also varying continuously with t, although the relative sizes of the eigenvalues will change with t at crossings where . We call these families homotopies of eigenfunctions. We study two explicit examples. The first example has Ω₀ equal to a square and Ω₁ equal to a circle; in both cases the eigenfunctions are known explicitly, so our homotopies connect these two explicit families. In the second example we approximate the Sierpinski carpet starting with a square, and we continuously delete subsquares of varying sizes. (Data available in full at ).

Keywords:

References

T. Tao , Poincaré's Legacies, Pages from Year Two of a Mathematical Blog. Part II ( American Mathematical Society , Providence, RI , 2009 ) . Google Scholar
Y. Colin de Verdière, Comment. Math. Helv. 63, 184 (1988), DOI: 10.1007/BF02566761. Crossref, Google Scholar
P. D. Lamberti and M. Lanza De Cristoforis, Math. Phys. Anal. Geom. 9, 65 (2006), DOI: 10.1007/s11040-005-9003-7. Crossref, Web of Science, Google Scholar
M. Teytel, Comm. Pure Appl. Math. 52, 917 (1999). Crossref, Web of Science, Google Scholar
T. Berry, S. Heilman and R. S. Strichartz, Exp. Math. 18(4), 449 (2009). Crossref, Web of Science, Google Scholar
M. T. Barlow and R. F. Bass, Ann. Ins. H. Poincaré 25, 225 (1989). Web of Science, Google Scholar
M. T. Barlow, R. F. Bass, T. Kumagai and A. Teplyaev, Uniqueness of Brownian motion on Sierpinski carpets, J. Eur. Math. Soc., to appear . Google Scholar
R. F. Bass, Trends in Probability and Related Analysis, Taipei, 1996 (World Scientific Publishing Co., River Edge, NJ, 1997) pp. 1–34. Google Scholar
T. Kato , Perturbation Theory for Linear Operators ( Springer-Verlag , New York , 1966 ) . Crossref, Google Scholar
A. Bäcker, R. Schubert and P. Stifter, J. Phys. A 30, 6783 (1997). Crossref, Google Scholar
P. Sarnak , Arithmetic quantum chaos. The Schur lectures (1992) (Tel Aviv), 183 , Israel Math. Conf. Proc. 8 ( Bar-Ilan Univ. , Ramat Gan , 1995 ) . Google Scholar
R. S. Strichartz , Differential Equations on Fractals: A Tutorial ( Princeton University Press , Princeton , 2006 ) . Crossref, Google Scholar
A. Hassell, Ann. Math. (2) 171, 605 (2010). Crossref, Web of Science, Google Scholar