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.
@article{JEDP_2015____A9_0,
author = {Sogge, Christopher D.},
title = {Problems related to the concentration of eigenfunctions},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
year = {2015},
pages = {1-11},
doi = {10.5802/jedp.638},
language = {en},
url = {http://dml.mathdoc.fr/item/JEDP_2015____A9_0}
}
Sogge, Christopher D. Problems related to the concentration of eigenfunctions. Journées équations aux dérivées partielles, (2015), pp. 1-11. doi : 10.5802/jedp.638. http://gdmltest.u-ga.fr/item/JEDP_2015____A9_0/
[1] On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., Tome 155 (1977) no. 3, pp. 249-276 | MR 455055 | Zbl 0341.35052
[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), Tome 17 (2015) no. 10, pp. 2513-2543 | MR 3420515
[5] Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions, Anal. PDE, Tome 8 (2015) no. 3, pp. 747-764 | Article | MR 3353830
[6] Geodesic restrictions and -estimates for eigenfunctions of Riemannian surfaces, Linear and complex analysis, Amer. Math. Soc., Providence, RI (Amer. Math. Soc. Transl. Ser. 2) Tome 226 (2009), pp. 27-35 | MR 2500507 | Zbl 1189.58015
[7] Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., Tome 138 (2007) no. 3, pp. 445-486 | Article | MR 2322684 | Zbl 1131.35053
[8] A few endpoint geodesic restriction estimates for eigenfunctions, Comm. Math. Phys., Tome 329 (2014) no. 2, pp. 435-459 | Article | MR 3210140 | Zbl 1293.58011
[9] Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., Tome 306 (2011) no. 3, pp. 777-784 | Article | MR 2825508 | Zbl 1238.58020
[10] Ergodicité et fonctions propres du laplacien, Comm. Math. Phys., Tome 102 (1985) no. 3, pp. 497-502 http://projecteuclid.org/euclid.cmp/1104114465 | MR 818831 | Zbl 0592.58050
[11] Small scale quantum ergodicity on negatively curved manifolds (2014) (http://arxiv.org/abs/1410.3911) | MR 3403398
[12] Improvement of eigenfunction estimates on manifolds of nonpositive curvature, Forum Math., Tome 27 (2015) no. 3, pp. 1435-1451 | MR 3341481
[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, Tome 5 (2012) no. 5, pp. 1133-1137 | Article | MR 3022851
[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., Tome 22 (1988) no. 3, p. 53-65, 96 | MR 961761 | Zbl 0679.35074
[17] Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, Tome 29 (1974) no. 6(180), p. 181-182 | MR 402834 | Zbl 0324.58020
[18] Localized -estimates of eigenfunctions: A note on an article of Hezari and Rivière (http://arxiv.org/abs/1503.07238) | MR 3439690
[19] Oscillatory integrals and spherical harmonics, Duke Math. J., Tome 53 (1986) no. 1, pp. 43-65 | Article | MR 835795 | Zbl 0636.42018
[20] Concerning the norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., Tome 77 (1988) no. 1, pp. 123-138 | Article | MR 930395 | Zbl 0641.46011
[21] Fourier integrals in classical analysis, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 105 (1993), pp. x+237 | Article | MR 1205579 | Zbl 0783.35001
[22] Kakeya-Nikodym averages and -norms of eigenfunctions, Tohoku Math. J. (2), Tome 63 (2011) no. 4, pp. 519-538 | Article | MR 2872954 | Zbl 1234.35156
[23] Hangzhou lectures on eigenfunctions of the Laplacian, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, Tome 188 (2014), pp. xii+193 | Article | MR 3186367
[24] About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., Tome 21 (2011) no. 1, pp. 150-173 | Article | MR 2755680 | Zbl 1214.58012
[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., Tome 114 (2002) no. 3, pp. 387-437 | Article | MR 1924569 | Zbl 1018.58010
[28] Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., Tome 18 (2011) no. 1, pp. 25-37 | Article | MR 2770580 | Zbl 1242.58017
[29] Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett., Tome 19 (2012) no. 6, pp. 1361-1364 | Article | MR 3091613 | Zbl 1283.58020
[30] On eigenfunction restriction estimates and -bounds for compact surfaces with nonpositive curvature, Advances in analysis: the legacy of Elias M. Stein, Princeton Univ. Press, Princeton, NJ (Princeton Math. Ser.) Tome 50 (2014), pp. 447-461 | MR 3329861
[31] Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J., Tome 55 (1987) no. 4, pp. 919-941 | Article | MR 916129 | Zbl 0643.58029
[32] On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., Tome 160 (1994) no. 1, pp. 81-92 http://projecteuclid.org/euclid.cmp/1104269516 | MR 1262192 | Zbl 0788.58043