Semiclassical Limits of Eigenfunctions on Flat n-Dimensional Tori

Aϊssiou, Tayeb

Canadian mathematical bulletin, Tome 56 (2013) no. 1, pp. 3-12

Résumé

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$ -dimensional flat torus ${{\mathbb{T}}^{n}}$ , and the Fourier transform of squares of the eigenfunctions ${{\left| \varphi \lambda\right|}^{2}}$ of the Laplacian have uniform ${{l}^{n}}$ bounds that do not depend on the eigenvalue $\lambda $ . The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on ${{\mathbb{T}}^{n+2}}$ . We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$ -dimensional sphere ${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius $\sqrt{\lambda }$ , and we use it in the proof.

Détail

DOI : 10.4153/CMB-2011-152-9

Mots-clés : 58G25, 81Q50, 35P20, 42B05, semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limits

Bibliographie

[1] [1] Avakumović, V.. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z. 65 (1956), 327–344. Google Scholar | DOI

[2] [2] Aϊssiou, T., Semiclassical limits of eigenforms and eigenfunctions on n-dimensional Tori. M.Sc. Thesis. McGill University, Montreal, QC, 2009. Google Scholar

[3] [3] Bourgain, J, Eigenfunction bounds for the Laplacian on the n-torus. Internat. Math. Res. Notices 1993, no. 3, 61–66. Google Scholar

[4] [4] Bourgain, J and Rudnick, Z, Restrictions of toral eigenfunctions to hypersurfaces. C. R. Math. Acad. Sci. Paris, 347(2009), no. 21-22, 1249–1253. Google Scholar

[5] [5] Bourgain, J and Rudnick, Z, On the nodal sets of toral eigenfunctions. To appear in Invent. Math. ArXiv:1003.1743. Google Scholar

[6] [6] Connes, B, Sur les coefficients des séries trigonométriques convergents sphériquement. C. R. Acad. Sci. Paris Sér. A-B 283(1976), no. 4, 159–161. Google Scholar

[7] [7] Duistermaat, J. J. and Guillemin, V.W., The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975), no. 1, 39–79. Google Scholar | DOI

[8] [8] Hörmander, L., The analysis of linear partial differential operators. IV. Springer-Verlag Berlin Heidelberg, 1985. Google Scholar

[9] [9] Hörmander, L., The spectral function of an elliptic operator. Acta. Math. 121 (1968), 193–218, Google Scholar | DOI

[10] [10] Jakobson, D., Quantum limits on flat tori. Ann. of Math. 145 (1997), no. 2, 235–266. Google Scholar | DOI

[11] [11] Jakobson, D, Nadirashvili, N, and Toth, J., Geometric properties of eigenfunctions. Russian Math. Surveys 56(2001), no. 6, 1085–1106. Google Scholar

[12] [12] Levitan, B. M. On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. Izvvestiya Akad. Nauk SSSR Ser. Mat. 16 (1952), 325–352 Google Scholar

[13] [13] Mockenhaupt, G. Bounds in Lebesgue spaces of oscillatory integral operators. Habilitationsschrift, Univ. Siegen, Siegen 1996. Google Scholar

[14] [14] Safarov, Y, Asymptotics of the spectrum of a pseudodifferential operator with periodic bicharacteristics. J. Soviet Math. 40 (1988), no. 5, 645–652. Google Scholar

[15] [15] Safarov, Y, Asymptotic of the spectral function of a positive elliptic operator without the nontrap condition. Funct. Anal. App. 22 (1988), no. 3, 213–223. Google Scholar | DOI

[16] [16] Safarov, Y and Vassiliev, D, The asymptotic distribution of eigenvalues of partial differential operators. Translations of Mathematical Monographs 155. American Mathematical Society, Providence, RI, 1997. Google Scholar

[17] [17] Sogge, C. Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact anifolds. J. Funct. Anal. 77 (1988), 123–138. Google Scholar | DOI

[18] [18] Sogge, C, Toth, J, and Zelditch, S., About the blowup of quasimodes on Riemannian manifolds. J. Geom. Anal. 21 (2011), no. 1, 150–173. Google Scholar | DOI

[19] [19] Sogge, C. and Zelditch, S., Riemannian manifolds with maximal eigenfunction growth. Duke Math. J. 114 (2002), no. 3, 387–437. Google Scholar | DOI

[20] [20] Toth, J. and Zelditch, S., Riemannian manifolds with uniformly bounded eigenfunctions. Duke Math. J. 111 (2002), no. 1, 97–132. Google Scholar | DOI

[21] [21] Zygmund, A., On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 (1974), 189–201. Google Scholar