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Journées équations aux dérivées partielles

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Semyon Dyatlov
Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
Journées équations aux dérivées partielles (2017), Exp. No. 4, 14 p., doi: 10.5802/jedp.654
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Résumé - Abstract

This expository article, written for the proceedings of the Journées EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [BD16] and Long Jin [DJ17]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.

Bibliographie

[An08] Nalini Anantharaman, Entropy and the localization of eigenfunctions, Ann. of Math. 168(2008), 435–475.
[AN07] Nalini Anantharaman and Stéphane Nonnenmacher, Half-delocalization of eigenfunctions of the Laplacian on an Anosov manifold, Ann. Inst. Fourier 57(2007), 2465–2523.
[BM62] Arne Beurling and Paul Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107(1962), 291–309.
[Bo16] David Borthwick, Spectral theory of infinite-area hyperbolic surfaces, second edition, Birkhäuser, 2016.
[BD16] Jean Bourgain and Semyon Dyatlov, Spectral gaps without the pressure condition, Ann. of Math., to appear.
[CdV85] Yves Colin de Verdière, Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys. 102(1985), 497–502.
[Dy17] Semyon Dyatlov, Notes on fractal uncertainty principle, lecture notes in progress, http://math.mit.edu/~dyatlov/files/2017/fupnotes.pdf.
[DJ17] Semyon Dyatlov and Long Jin, Semiclassical measures on hyperbolic surfaces have full support, preprint, arXiv:1705.05019.
[DZ16] Semyon Dyatlov and Joshua Zahl, Spectral gaps, additive energy, and a fractal uncertainty principle, Geom. Funct. Anal. 26(2016), 1011–1094.
[Ji17] Long Jin, Control for Schrödinger equation on hyperbolic surfaces, preprint, arXiv:1707.04990.
[JZ17] Long Jin and Ruixiang Zhang, Fractal uncertainty principle with explicit exponent, preprint, arXiv:1710.00250.
[Li06] Elon Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. 163(2006), 165–219.
[Ma06] Jens Marklof, Arithmetic quantum chaos, Encyclopedia of Mathematical Physics, Oxford: Elsevier 1(2006), 212–220.
[RS94] Zeév Rudnick and Peter Sarnak, The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161(1994), 195–213.
[Sa11] Peter Sarnak, Recent progress on the quantum unique ergodicity conjecture, Bull. Amer. Math. Soc. 48(2011), 211–228.
[Sh74] Alexander Shnirelman, Ergodic properties of eigenfunctions, Usp. Mat. Nauk. 29(1974), 181–182.
[Ze87] Steve Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55(1987), 919–941.
[Ze09] Steve Zelditch, Recent developments in mathematical quantum chaos, Curr. Dev. Math. 2009, 115–204.
[Zw12] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, AMS, 2012.
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