The Experts below are selected from a list of 1908 Experts worldwide ranked by ideXlab platform
Dyatlov Semyon - One of the best experts on this subject based on the ideXlab platform.
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Control of eigenfunctions on surfaces of variable curvature
2021Co-Authors: Dyatlov Semyon, Jin Long, Nonnenmacher StéphaneAbstract:We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schr\"odinger equation by any Nonempty Open Set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works [arXiv:1705.05019], [arXiv:1712.02692], which considered the Setting of surfaces of constant negative curvature. The proofs use the strategy of [arXiv:1705.05019], [arXiv:1712.02692] and rely on the fractal uncertainty principle of [arXiv:1612.09040]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used in [arXiv:1504.06589]. Instead, our argument uses Egorov's Theorem up to local Ehrenfest time and the hyperbolic parametrix of [arXiv:0706.3242], together with the $C^{1+}$ regularity of the stable/unstable foliations.Comment: 119+eps pages, 13 figures. To appear in J. Amer. Math. So
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Lower bounds on eigenfunctions and fractal uncertainty principle
Banff International Research Station for Mathematical Innovation and Discovery, 2018Co-Authors: Dyatlov SemyonAbstract:Let $(M,g)$ be a compact Riemannian manifold and $\Omega\subSet M$ a Nonempty Open Set. Take an $L^2$ normalized eigenfunction $u$ of the Laplacian on $M$ with eigenvalue $\lambda^2$. What lower bounds can we get on the mass $m_\Omega(u)=\int_\Omega |u|^2$ There are two well-known bounds for general $M$: (a) $m_\Omega(u)\geq ce^{-C\lambda}$, following from unique continuation estimates, and (b) $m_\Omega(u)\geq c$, where $c>0$ is independent of $\lambda$, assuming that $\Omega$ intersects every sufficiently long geodesic (this is known as the geometric control condition). In general one cannot improve on the bound (a) for arbitrary $\Omega$, as illustrated by Gaussian beams on the round sphere. I will present a recent result which establishes the frequency-independent lower bound (b) for any choice of $\Omega$ when $M$ is a surface of constant negative curvature. This bound has numerous applications, such as control for the Schrödinger equation, exponential decay of damped waves, and the full support property of semiclassical measures. The proof uses the chaotic nature of the geodesic flow on $M$. The key new ingredient is a recently established fractal uncertainty principle, which states that no function can be localized close to a fractal Set in both position and frequency. This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.Non UBCUnreviewedAuthor affiliation: University of BerkeleyFacult
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Lower bounds on eigenfunctions on hyperbolic surfaces.
Banff International Research Station for Mathematical Innovation and Discovery, 2018Co-Authors: Dyatlov SemyonAbstract:I show that on a compact hyperbolic surface, the mass of an L2- normalized eigenfunction of the Laplacian on any Nonempty Open Set is bounded below by a positive constant depending on the Set, but not on the eigenvalue. This statement, more precisely its stronger semiclassi- cal version, has many applications including control for the Schrdinger equation and the full support property for semiclassical defect mea- sures. The key new ingredient of the proof is a fractal uncertainty principle, stating that no function can be localized close to a porous Set in both position and frequency. This talk is based on joint works with Long Jin and with Jean Bourgain.Non UBCUnreviewedAuthor affiliation: MassachuSetts Institute of TechnologyFacult
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Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle
2017Co-Authors: Dyatlov SemyonAbstract:This expository article, written for the proceedings of the Journ\'ees EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. 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.Comment: 18 page
Wu Weisheng - One of the best experts on this subject based on the ideXlab platform.
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MODIFIED SCHMIDT GAMES AND NON-DENSE FORWARD ORBITS OF PARTIALLY HYPERBOLIC SYSTEMS
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS, 2016Co-Authors: Wu WeishengAbstract:Let f : M -> M be a C1+theta-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by f and played on any unstable manifold. Utilizing it we generalize some results of [25] as follows. Consider a Set of points with non-dense forward orbit: E(f, y) := {z is an element of M : y is not an element of<({f(k)(z), k is an element of N}})over bar> for some y is an element of M and E-x(f, y) := E(f, y) boolean AND W-u(x) for any x is an element of M. We show that E-x (f, y) is a winning Set for such modified Schmidt games played on W-u(x), which implies that E-x(f,y) has Hausdorff dimension equal to dim W-u(x). Then for any Nonempty Open Set V subSet of M we show that E(f, y) boolean AND v has full Hausdorff dimension equal to dim M, by using a technique of constructing measures supported on E(f, y) with lower pointwise dimension approximating dim M.SCI(E)ARTICLEwuweisheng@math.pku.edu.cn63463-34813
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Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems
2015Co-Authors: Wu WeishengAbstract:Let $f: M \to M$ be a $C^{1+\theta}$-partially hyperbolic diffeomorphism. We introduce a type of modified Schmidt games which is induced by $f$ and played on any unstable manifold. Utilizing it we generalize some results of \cite{Wu} as follows. Consider a Set of points with non-dense forward orbit: $$E(f, y) := \{ z\in M: y\notin \overline{\{f^k(z), k \in \mathbb{N}\}}\}$$ for some $y \in M$ and $$E_{x}(f, y) := E(f, y) \cap W^u(x)$$ for any $x\in M$. We show that $E_x(f,y)$ is a winning Set for such modified Schmidt games played on $W^u(x)$, which implies that $E_x(f,y)$ has Hausdorff dimension equal to $\dim W^u(x)$. Then for any Nonempty Open Set $V \subSet M$ we show that $E(f, y) \cap V$ has full Hausdorff dimension equal to $\dim M$, by using a technique of constructing measures supported on $E(f, y)$ with lower pointwise dimension approximating $\dim M$.Comment: 19 pages. Remark 4.10 is corrected. We have followed the proof scheme in \cite{Wu}. arXiv admin note: text overlap with arXiv:1311.530
Jin Long - One of the best experts on this subject based on the ideXlab platform.
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Control of eigenfunctions on surfaces of variable curvature
2021Co-Authors: Dyatlov Semyon, Jin Long, Nonnenmacher StéphaneAbstract:We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schr\"odinger equation by any Nonempty Open Set, and shows that every semiclassical measure has full support. We also prove exponential energy decay for solutions to the damped wave equation on such surfaces, for any nontrivial damping coefficient. These results extend previous works [arXiv:1705.05019], [arXiv:1712.02692], which considered the Setting of surfaces of constant negative curvature. The proofs use the strategy of [arXiv:1705.05019], [arXiv:1712.02692] and rely on the fractal uncertainty principle of [arXiv:1612.09040]. However, in the variable curvature case the stable/unstable foliations are not smooth, so we can no longer associate to these foliations a pseudodifferential calculus of the type used in [arXiv:1504.06589]. Instead, our argument uses Egorov's Theorem up to local Ehrenfest time and the hyperbolic parametrix of [arXiv:0706.3242], together with the $C^{1+}$ regularity of the stable/unstable foliations.Comment: 119+eps pages, 13 figures. To appear in J. Amer. Math. So
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Control for Schr\"odinger equation on hyperbolic surfaces
2018Co-Authors: Jin LongAbstract:We show that the any Nonempty Open Set on a hyperbolic surface provides observability and control for the time dependent Schr\"odinger equation. The only other manifolds for which this was previously known are flat tori. The proof is based on the main estimate in Dyatlov-Jin and standard arguments of control theory.Comment: 10 page
Léautaud Matthieu - One of the best experts on this subject based on the ideXlab platform.
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Tunneling estimates and approximate controllability for hypoelliptic equations
2017Co-Authors: Laurent Camille, Léautaud MatthieuAbstract:This article is concerned with quantitative unique continuation estimates for equations involving a "sum of squares" operator $\mathcal{L}$ on a compact manifold $\mathcal{M}$ assuming: $(i)$ the Chow-Rashevski-H\"ormander condition ensuring the hypoellipticity of $\mathcal{L}$, and $(ii)$ the analyticity of $\mathcal{M}$ and the coefficients of $\mathcal{L}$. The first result is the tunneling estimate $\|\varphi\|_{L^2(\omega)} \geq Ce^{- \lambda^{\frac{k}{2}}}$ for normalized eigenfunctions $\varphi$ of $\mathcal{L}$ from a Nonempty Open Set $\omega\subSet \mathcal{M}$, where $k$ is the hypoellipticity index of $\mathcal{L}$ and $\lambda$ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation $(\partial_t^2+\mathcal{L})u=0$: for $T>2 \sup_{x \in \mathcal{M}}(dist(x,\omega))$ (here, $dist$ is the sub-Riemannian distance), the observation of the solution on $(0,T)\times \omega$ determines the data. The constant involved in the estimate is $Ce^{c\Lambda^k}$ where $\Lambda$ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation $(\partial_t+\mathcal{L})v=1_\omega f$ in any time, with appropriate (exponential) cost, depending on $k$. In case $k=2$ (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary $\partial \mathcal{M}$ can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in arxiv:1506.04254
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Wigner measures and observability for the Schr\"odinger equation on the disk
2015Co-Authors: Anantharaman Nalini, Léautaud Matthieu, Macià FabricioAbstract:We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schr\"{o}dinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a "structure theorem", expressing the restriction of the Wigner measures on each invariant torus in terms of {\em second-microlocal measures}. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $L^2$-norm of a solution on any Open subSet intersecting the boundary (resp. the $L^2$-norm of the Neumann trace on any Nonempty Open Set of the boundary) controls its full $L^2$-norm (resp. $H^1$-norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.Comment: We modified the introduction but not the content of the articl
Matthieu Léautaud - One of the best experts on this subject based on the ideXlab platform.
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Tunneling estimates and approximate controllability for hypoelliptic equations
2018Co-Authors: Camille Laurent, Matthieu LéautaudAbstract:This article is concerned with quantitative unique continuation estimates for equations involving a " sum of squares " operator L on a compact manifold M assuming: (i) the Chow-Rashevski-Hörmander condition ensuring the hypoellipticity of L, and (ii) the analyticity of M and the coefficients of L. The first result is the tunneling estimate ϕ L 2 (ω) ≥ Ce −λ k 2 for normalized eigenfunctions ϕ of L from a Nonempty Open Set ω ⊂ M, where k is the hypoellipticity index of L and λ the eigenvalue. The main result is a stability estimate for solutions to the hypoelliptic wave equation (∂ 2 t +L)u = 0: for T > 2 sup x∈M (dist(x, ω)) (here, dist is the sub-Riemannian distance), the observation of the solution on (0, T) × ω determines the data. The constant involved in the estimate is Ce cΛ k where Λ is the typical frequency of the data. We then prove the approximate controllability of the hypoelliptic heat equation (∂t + L)v = 1ωf in any time, with appropriate (exponential) cost, depending on k. In case k = 2 (Grushin, Heisenberg...), we further show approximate controllability to trajectories with polynomial cost in large time. We also explain how the analyticity assumption can be relaxed, and a boundary ∂M can be added in some situations. Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy developed by the authors in [LL15].
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2016Co-Authors: Nalini Anantharaman, Matthieu Léautaud, Fabricio MaciàAbstract:Abstract. We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of two-microlocal measures, propagating according to Schrödinger equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the L2-norm of a solution on any Open subSet intersecting the boundary (resp. the L2-norm of the Neumann trace on any Nonempty Open Set of the boundary) controls its full L2-norm (resp. H1-norm). These results show in particular that the energy of solutions cannot concentrate on periodi