We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for -norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.
Keywords: Eigenfunctions, Kakeya-Nikodym averages
@incollection{JEDP_2015____A9_0,
author = {Christopher D. Sogge},
title = {Problems related to the concentration of eigenfunctions},
booktitle = {},
series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {9},
pages = {1--11},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2015},
doi = {10.5802/jedp.638},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.638/}
}
TY - JOUR AU - Christopher D. Sogge TI - Problems related to the concentration of eigenfunctions JO - Journées équations aux dérivées partielles PY - 2015 SP - 1 EP - 11 PB - Groupement de recherche 2434 du CNRS UR - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.638/ DO - 10.5802/jedp.638 LA - en ID - JEDP_2015____A9_0 ER -
%0 Journal Article %A Christopher D. Sogge %T Problems related to the concentration of eigenfunctions %J Journées équations aux dérivées partielles %D 2015 %P 1-11 %I Groupement de recherche 2434 du CNRS %U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.638/ %R 10.5802/jedp.638 %G en %F JEDP_2015____A9_0
Christopher D. Sogge. Problems related to the concentration of eigenfunctions. Journées équations aux dérivées partielles (2015), article no. 9, 11 p. doi : 10.5802/jedp.638. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.638/
[1] On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Volume 155 (1977) no. 3, pp. 249-276 | MR | Zbl
[2] Concerning Topogonov’s Theorem and logarithmic improvement of estimates of eigenfunctions (preprint)
[3] Refined and microlocal Kakeya-Nikodym bounds of eigenfunctions in higher dimensions (preprint)
[4] On Kakeya–Nikodym averages, -norms and lower bounds for nodal sets of eigenfunctions in higher dimensions, J. Eur. Math. Soc. (JEMS), Volume 17 (2015) no. 10, pp. 2513-2543 | MR
[5] Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions, Anal. PDE, Volume 8 (2015) no. 3, pp. 747-764 | DOI | MR
[6] Geodesic restrictions and -estimates for eigenfunctions of Riemannian surfaces, Linear and complex analysis (Amer. Math. Soc. Transl. Ser. 2), Volume 226, Amer. Math. Soc., Providence, RI, 2009, pp. 27-35 | MR | Zbl
[7] Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., Volume 138 (2007) no. 3, pp. 445-486 | DOI | MR | Zbl
[8] A few endpoint geodesic restriction estimates for eigenfunctions, Comm. Math. Phys., Volume 329 (2014) no. 2, pp. 435-459 | DOI | MR | Zbl
[9] Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., Volume 306 (2011) no. 3, pp. 777-784 | DOI | MR | Zbl
[10] Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Volume 102 (1985) no. 3, pp. 497-502 http://projecteuclid.org/euclid.cmp/1104114465 | MR | Zbl
[11] Small scale quantum ergodicity on negatively curved manifolds (2014) (http://arxiv.org/abs/1410.3911) | MR
[12] Improvement of eigenfunction estimates on manifolds of nonpositive curvature, Forum Math., Volume 27 (2015) no. 3, pp. 1435-1451 | MR
[13] norms, nodal sets, and quantum ergodicity (2014) (http://arxiv.org/abs/1411.4078)
[14] A natural lower bound for the size of nodal sets, Anal. PDE, Volume 5 (2012) no. 5, pp. 1133-1137 | DOI | MR
[15] Small scale equidistribution of eigenfunctions on the torus (http://arxiv.org/abs/1508.01074)
[16] Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition, Funktsional. Anal. i Prilozhen., Volume 22 (1988) no. 3, p. 53-65, 96 | MR | Zbl
[17] Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Volume 29 (1974) no. 6(180), pp. 181-182 | MR | Zbl
[18] Localized -estimates of eigenfunctions: A note on an article of Hezari and Rivière (http://arxiv.org/abs/1503.07238) | MR
[19] Oscillatory integrals and spherical harmonics, Duke Math. J., Volume 53 (1986) no. 1, pp. 43-65 | DOI | MR | Zbl
[20] Concerning the norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Volume 77 (1988) no. 1, pp. 123-138 | DOI | MR | Zbl
[21] Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, 105, Cambridge University Press, Cambridge, 1993, pp. x+237 | DOI | MR | Zbl
[22] Kakeya-Nikodym averages and -norms of eigenfunctions, Tohoku Math. J. (2), Volume 63 (2011) no. 4, pp. 519-538 | DOI | MR | Zbl
[23] Hangzhou lectures on eigenfunctions of the Laplacian, Annals of Mathematics Studies, 188, Princeton University Press, Princeton, NJ, 2014, pp. xii+193 | DOI | MR
[24] About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., Volume 21 (2011) no. 1, pp. 150-173 | DOI | MR | Zbl
[25] Focal points and sup-norms of eigenfunctions (Rev. Mat. Iberoamericana, to appear.)
[26] Focal points and sup-norms of eigenfunctions manifolds II: the two-dimensional case (Rev. Mat. Iberoamericana, to appear.)
[27] Riemannian manifolds with maximal eigenfunction growth, Duke Math. J., Volume 114 (2002) no. 3, pp. 387-437 | DOI | MR | Zbl
[28] Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., Volume 18 (2011) no. 1, pp. 25-37 | DOI | MR | Zbl
[29] Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett., Volume 19 (2012) no. 6, pp. 1361-1364 | DOI | MR | Zbl
[30] On eigenfunction restriction estimates and -bounds for compact surfaces with nonpositive curvature, Advances in analysis: the legacy of Elias M. Stein (Princeton Math. Ser.), Volume 50, Princeton Univ. Press, Princeton, NJ, 2014, pp. 447-461 | MR
[31] Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Volume 55 (1987) no. 4, pp. 919-941 | DOI | MR | Zbl
[32] On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Volume 160 (1994) no. 1, pp. 81-92 http://projecteuclid.org/euclid.cmp/1104269516 | MR | Zbl
Cité par Sources :