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Semiclassical measures on hyperbolic surfaces have full support

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We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated in “Spectral gaps, additive energy, and a fractal uncertainty principle” [Dyatlov, S. & Zahl, J. Geom. Funct. Anal., 26 (2016), 1011–1094] and proved for porous sets in “Spectral gaps without the pressure condition” [Bourgain, J. & Dyatlov, S. Ann. of Math., 187 (2018), 825–867].

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... Nevertheless there exist flow-invariant measures with high entropy which are supported on invariant proper subsets. A recent result due to Dyatlov and Jin , [DJ18], states that for every open ∅ = Ω ⊂ M there exists a constant c Ω > 0 such that for every eigenmode ϕ j Ω |ϕ j | 2 dx ≥ c Ω M |ϕ j | 2 dx = c Ω . ...
... Our paper is dedicated to proving a result analogous to [DJ18] for the quantum cat map, that every semi-classical measure µ sc associated with γ is fully supported on T 2 meaning that all sequences of eigenstates {ϕ N } N are fully delocalized on T 2 for N large enough, Theorem 1.1. Let γ ∈Γ (2) (whereΓ (2) is defined in (4)) be a hyperbolic matrix quantized into the family {M N (γ)} N . ...
... Then, we invoke a combinatorial argument from [DJ18], Lemma 3.7 (Lemma 3.3 in [DJ18]). There is some C (that might depend on δ) such that for every h, #X ≤ Ch −4 √ δ . ...
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We consider the quantum cat map - a toy model of a quantized chaotic system. We show that its eigenstates are fully delocalized on $\mathbb{T}^2$ in the semiclassical limit (or equivalently that each semiclassical measure is fully supported on $\mathbb{T}^2$). We adapt the proof of a similar result proved for the eigenstates of $-\Delta_g$ on compact hyperbolic surfaces from [arXiv:1705.05019], relying on the fractal uncertainty principle in [arXiv:1612.09040].
... The purpose of this expository article is to present a recent result of Dyatlov-Jin [DJ17] on control of eigenfunctions on hyperbolic surfaces. The main tool is the fractal uncertainty principle proved by Bourgain-Dyatlov [BD16]. ...
... The main tool is the fractal uncertainty principle proved by Bourgain-Dyatlov [BD16]. To keep the article short we will omit many technical details of the proofs, referring the reader to the original papers [BD16,DJ17] and to the lecture notes [Dy17]. The results discussed here belong to quantum chaos; see [Ma06,Ze09,Sa11] for introductions to the subject. ...
... The question of which ϕ t -invariant probability measures on S * M can be semiclassical measures has received considerable attention in the quantum chaos literature. Here we only mention a few works, referring the reader to the introduction to [DJ17] for a more comprehensive overview. ...
Article
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.
... We would like to mention an outstanding recent result from the works by Bourgain & Dyatlov [3] and Dyatlov & Jin [7]. ...
... In particular, for negatively curved surfaces the quantum unique ergodicity conjecture states that asymptotically the L 2 mass of eigenfunctions is distrubed uniformly. We refer to [26], [15], [28], [31], [32], [33], [7] for the results on ergodic properties of eigenfunctions. ...
... Theorem 1.1 ([3],[7]). Under assumption that (M, g) is a closed Riemannian surface with constant negative curvature the following inequality holdsfor Laplace eigenfunctions on M . ...
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Let $u$ be a solution to an elliptic equation $\text{div}(A\nabla u)=0$ with Lipschitz coefficients in $\mathbb{R}^n$. Assume $|u|$ is bounded by $1$ in the ball $B=\{|x|\leq 1\}$. We show that if $|u| < \varepsilon$ on a set $ E \subset \frac{1}{2} B$ with positive $n$-dimensional Hausdorf measure, then $$|u|\leq C\varepsilon^\gamma \text{ on } \frac{1}{2}B,$$ where $C>0, \gamma \in (0,1)$ do not depend on $u$ and depend only on $A$ and the measure of $E$. We specify the dependence on the measure of $E$ in the form of the Remez type inequality. Similar estimate holds for sets $E$ with Hausdorff dimension bigger than $n-1$. For the gradients of the solutions we show that a similar propagation of smallness holds for sets of Hausdorff dimension bigger than $n-1-c$, where $c>0$ is a small numerical constant depending on the dimension only.
... In our paper, we adapt the approach from a joint work with Dyatlov [DyJi18] which shows that semiclassical measures for Laplacian eigenfunctions on compact hyperbolic surfaces have full support. The key idea in [DyJi18] is a new approach called fractal uncertainty principle developed by Dyatlov-Zahl [DyZa16] and Bourgain-Dyatlov [BoDy18] to obtain essential spectral gaps for convex co-compact hyperbolic surfaces when the pressure condition fails. ...
... In our paper, we adapt the approach from a joint work with Dyatlov [DyJi18] which shows that semiclassical measures for Laplacian eigenfunctions on compact hyperbolic surfaces have full support. The key idea in [DyJi18] is a new approach called fractal uncertainty principle developed by Dyatlov-Zahl [DyZa16] and Bourgain-Dyatlov [BoDy18] to obtain essential spectral gaps for convex co-compact hyperbolic surfaces when the pressure condition fails. A basic version of fractal uncertainty principle states that a function and its Fourier transform cannot be both localized near fractal sets. ...
... 2, we review some basic facts about hyperbolic surfaces and semiclassical analysis, especially the exotic symbol calculus (Sect. 2.3) developped in [DyZa16] and [DyJi18]. Then we formulate Theorem 2.2 about the decay of a general semiclassical damped propagator localizing near the energy surface (defined in (2.22)). ...
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We prove exponential decay of energy for solutions of the damped wave equation on compact hyperbolic surfaces with regular initial data as long as the damping is nontrivial. The proof is based on a similar strategy as in Dyatlov-Jin and in particular, uses the fractal uncertainty principle proved in Bourgain-Dyatlov.
... Determining the range of validity (in terms of T ) of this kind of approximation is a key issue when studying the fine properties of Laplace eigenfunctions [16,137,2,8,6,49]. In general, the best one can expect is that such a correspondence remains true up to times of order c| log | with c > 0 which can be expressed in terms of certain expansion rates of the geodesic flow [14,27]. ...
... In the case where ω is an open set of M, these results implies an observability Theorem like Theorem 5.3 provided that P top Λ S * ω , 1 2 < 0. In the case of hyperbolic surfaces, this condition was removed by Dyatlov and Jin [49] using a fractal uncertainty principle due to Bourgain and Dyatlov [22] -see also [51]: Compared with the case of flat tori, we emphasize that this Theorem is valid for any nonempty open set inside S * M (and not only on M). An extension of this result to variable negative curvature was recently announced by Dyatlov, Jin and Nonnenmacher [50]. ...
... If this fraction is small enough in terms of the exponent β > 0 appearing in the fractal uncertainty principle (55), then one can still apply the estimate to these extra terms. The remaining sums are again much smaller and performing an argument similar to the one from [2] allows to remove the logarithmic factor -see [49] for details. ...
Preprint
Given a quantum Hamiltonian, we explain how the dynamical properties of the underlying classical system affect the behaviour of quantum eigenstates in the semi-classical limit. We study this problem via the notion of semiclassical measures. We mostly focus on two opposite dynamical paradigms: completely integrable systems and chaotic ones. We recall standard tools from microlocal analysis and from dynamical systems. We show how to use them in order to illustrate the classical-quantum correspondance and to compare properties of completely integrable and chaotic systems.
... 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 [DJ18,Ji17], which considered the setting of surfaces of constant negative curvature. ...
... Our Theorem 3 gives a different type of restriction on µ. As explained in [DJ18], there exist invariant measures which are excluded by Theorem 3 but not by entropy bounds, and vice versa. For instance, on any Anosov surface one can construct flow invariant fractal subsets F S * M of Hausdorff dimension close to 3, which support invariant measures of large entropy. ...
... In the special case of hyperbolic surfaces, Theorems 1-3 were proved by Dyatlov-Jin [DJ18]; see also the reviews [Dy17,Dy19]. The proofs in the present paper partially use the strategy of [DJ18], in particular they rely on the fractal uncertainty principle (FUP) established by Bourgain-Dyatlov [BD18]. ...
Preprint
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.
... The measure of maximum entroy is given by the Liouville measure, and thus "eigenfunctions are at least half delocalized". Dyatlov and Jin [10] consequently showed that any quantum limit must have full support in S * (M ), for compact hyperbolic surfaces M with constant negative curvature; together with Nonnenmacher this was recently strengthened [11] to the include the case of surfaces with variable negative curvature. ...
... The subsequence of almost primes {n ℓ } constructed as described above creates the imbalance in the spectral equation (1.2) by boosting the contribution of the terms m = Q 0 n ℓ , Q 0 n ℓ + 4. The next step in our argument is to show that this imbalance typically overwhelms the contribution of the remaining terms. To do this, we first show in Section 3 that for all new eigenvalues lying outside a small exceptional set the spectral equation (1.2) can be effectively truncated to integers m with essentially |m−λ| ≪ (log λ) 10 . This is done by controlling sums of r(n) over short intervals and uses a second moment estimate of the Dedekind zeta-function ζ Q(i) . ...
... |B(x; q, a, ε)| ≪ x (log x) 10 for Q ≤ x 1/2 /(log x) B 0 . ...
Preprint
We consider momentum push-forwards of measures arising as quantum limits (semi-classical measures) of eigenfunctions of a point scatterer on the standard flat torus $\mathbb T^2 = \mathbb R^2/\mathbb Z^{2}$. Given any probability measure arising by placing delta masses, with equal weights, on $\mathbb Z^2$-lattice points on circles and projecting to the unit circle, we show that the mass of certain subsequences of eigenfunctions, in momentum space, completely localizes on that measure and is completely delocalized in position (i.e., concentration on Lagrangian states.) We also show that the mass, in momentum, can fully localize on more exotic measures, e.g. singular continous ones with support on Cantor sets. Further, we can give examples of quantum limits that are certain convex combinations of such measures, in particular showing that the set of quantum limits is richer than the ones arising only from weak limits of lattice points on circles. The proofs exploit features of the half-dimensional sieve and behavior of multiplicative functions in short intervals.
... As mentioned in the beginning of the introduction, in the setting of negatively curved surfaces the paper [DJN19] gives another restriction: all semiclassical measures have full support. In the case of hyperbolic surfaces this was previously proved by Dyatlov-Jin [DJ18]. ...
... (1.15) To remove the log N prefactor, we revise the decomposition I = B X + B Y in the same way as in [DJ18,DJN19] (which in turn was inspired by [Ana08]), including more terms into B X and using that the norm bound in (1.14) (or rather, its slight generalization) is a negative power of N. See §3.1.2 for details. ...
... 2. If L is Lagrangian, then a version of the class S L,ρ,ρ corresponding to compactly supported symbols but an arbitrary (not necessarily constant) Lagrangian foliation previously appeared in [DJ18] which inspired part of the argument in the present paper. In the important special case ρ = 0 this class was introduced in [DZ16]. ...
Preprint
Consider a quantum cat map $M$ associated to a matrix $A\in\mathop{\mathrm{Sp}}(2n,\mathbb Z)$, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of $M$ on any nonempty open set in the position-frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of $A$ of largest absolute value and (2) the characteristic polynomial of $A$ is irreducible over the rationals. This is similar to previous work [arXiv:1705.05019], [arXiv:1906.08923] on negatively curved surfaces and [arXiv:2103.06633] on quantum cat maps with $n=1$, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.
... In a recent work of Dyatlov-Jin [7], the fractal uncertainty principle (Theorem 1.1) is also applied to closed quantum chaotic systems, to understand the behavior of Laplace eigenfunctions on compact hyperbolic surfaces in the semiclassical limit. In particular, it is shown that any semiclassical measure must have full support on the unit cosphere bundle. ...
... The Quantum Unique Ergodicity Conjecture of Rudnick-Sarnak [18] states that the Liouville measure is the only semiclassical measure. For any subset U of the unit cosphere bundle, combining Theorem 1.2 with the argument in [7] should give a lower bound on the semiclassical measure of U , in terms only of the geometry of U . But the bound obtained in this way seems unlikely to be even comparable to the Liouville measure. ...
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We prove an explicit formula for the dependence of the exponent β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} in the fractal uncertainty principle of Bourgain–Dyatlov (Ann Math 187:1–43, 2018) on the dimension δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} and on the regularity constant CR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_R$$\end{document} for the regular set. In particular, this implies an explicit essential spectral gap for convex co-compact hyperbolic surfaces when the Hausdorff dimension of the limit set is close to 1.
... We give two proofs of the result, by two rather different approaches whose contrast seems to be of some independent interest. Both are based on the Dyatlov-Jin(-Nonnenmacher) observability estimate [DJ17,DJN19]. The first proof is based on the Rellich identity of [CTZ13] (adapted from [Bu]) relating interior and restricted matrix elements and microlocal lifts of eigenfunctions. ...
... Then, lim h→0Op h (b 0 )u| H L 2 (H) + Op h (b 0 )h∂ ν u| H L 2 (H) = 0. In particular, by Lemma 4, there is a ∈ C ∞ c (T * M ) with a ∩ S * M = 0 such that lim h→0 Op h (a)u L 2 (M) = 0.But, this contradictions the results of[DJ17,DJN19]. ...
Preprint
We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.
... Remark 2.1. The recent works [2], [8], [9] on the distribution of L 2 mass of eigenfunctions imply that for two-dimensional surfaces with negative curvature, one can improve ...
... Let ρ = r2 −m . Using (8) and applying growth Lemma 2.5 for B 2 −j r (x i ) for j ∈ {0, 1, . . . , m}, we have (9) sup ...
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Let (M n , g) be a closed n-dimensional Riemannian mani-fold, where g = (gij) is C 1-smooth metric. Consider the sequence of eigenfunctions u k of the Laplace operator on M. Let B be a ball on M. We prove that the number of nodal domains of u k that intersect B is not greater than C1 Volumeg(B) Volumeg(M) k + C2k n−1 n , where C1, C2 depend on M. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.
... Remark 2.1. The recent works [2], [8], [9] on the distribution of L 2 mass of eigenfunctions imply that for two-dimensional surfaces with negative curvature, one can improve ...
... Let ρ = r2 −m . Using (8) and applying growth Lemma 2.5 for B 2 −j r (x i ) for j ∈ {0, 1, . . . , m}, we have (9) sup ...
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Let $(M, g)$ be a closed Riemannian manifold, where g is $C^1$-smooth metric. Consider the sequence of eigenfunctions $u_k$ of the Laplace operator on M. Let $B$ be a ball on $M$. We prove a sharp estimate of the number of nodal domains of $u_k$ that intersect $B$. The problem of local bounds for the volume and for the number of nodal domains was raised by Donnelly and Fefferman, who also proposed an idea how one can prove such bounds. We combine their idea with two ingredients: the recent sharp Remez type inequality for eigenfunctions and the Landis type growth lemma in narrow domains.
... In the context of quantum ergodicity (unique or not) on negatively curved surfaces, it is important to emphasize two relatively recent results that were tremendous steps in our understanding of high-frequency spectral asymptotics. The first was the breakthrough of Dyatlov-Jin [16], underpinned by an earlier breakthrough of Bourgain-Dyatlov [6] on resonance gaps for hyperbolic surfaces with funnels, proving that sequences of eigenfunctions on compact hyperbolic surfaces must delocalize across all of position space in the following sense: Supp µ sc = E. This was later generalized to surfaces of negative sectional curvatures by Dyatlov-Jin-Nonnenmacher [17]. ...
... The works [8,19] are particular to constant negative curvature with the work of Eswarathasan-Silberman holding in higher dimensions. We emphasize that the results of Dyatlov-Jin-Nonnenmacher [16,17] on the full support of µ sc hold for quasimodes of width o h | log | . We are now in a position to introduce our main objects of study: Definition 1.1.5 ...
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Consider $(M,g)$ a compact, boundaryless Riemannian manifold admitting an Anosov geodesic flow. Let $\epsilon < \min\{1, \frac{\lambda_{\max}}{2}\}$ where $\lambda_{\max}$ is the maximal expansion rate for $(M,g)$. We study the semiclassical measures $\mu_{sc}$ of $\epsilon$-logarithmic modes, which are quasimodes spectrally supported in intervals of width $\epsilon \frac{\hbar}{|\log \hbar|}$, of the Laplace-Beltrami operator on $M$. Under a technical assumption on the support of a signed measure related to $\mu_{sc}$, we show that the lower bound for the Kolmogorov-Sinai entropy of $\mu_{sc}$ generalizes that of Ananthamaran-Koch-Nonnenmacher. A feature of our results is the quantitative nature of our bounds on hyperbolic manifolds. Via previous results of the author with Nonnenmacher and Silberman, we provide another demonstration of the existence of $\epsilon$-logarithmic modes whose semiclassical measures possess zero entropy ergodic components yet have other components with positive entropy.
... To verify QUE, one needs to show that functions u (s) cannot have microlocal singularities in the semiclassical limit. In this direction, some significant results on deconcentration of eigenfunctions for the general non-arithmetic case were obtained in [An08], [AnNo07] and [DJ17]. ...
... Excitation evolution therefore should be understood as quantum homotopy between quantum magnetic systems, this homotopy is quantization of homotopy between classicalτ -hypercyclic flows on X. We've got one more implementation of Bohr principle. Recall that the latter states various kinds of correspondence between classical dynamical systems and their quantizations in the semiclassical limit → 0. Now, suppose that 0.01 of mass of some weak* limit of sequence {(u (s) 0 ) 2 · A} s∈J turned to be concentrated on a closed geodesic loop γ (to author's best knowledge, such a possibility still is not disproven, at least, it is not prohibited by [An08], [AnNo07] and [DJ17]); then 0.01 of mass of weak* limit of the corresponding sequence u (s) τ 2 · A s∈J will be concentrated on the twoτ -hypercycles obtained by shifting γ by distance ln τ 2 + √τ 2 + 1 in the left-and right-normal directions. Ifτ → +∞ then both theseτ -hypercycles become long, close to horocycles and thence almost uniformly distributed in X and in S 1 X. ...
Preprint
What will be if, given a pure stationary state on a compact hyperbolic surface, we start applying creation operator every $\hbar$ "adiabatic" second? It turns that during adiabatic time comparable to 1 wavefunction will change as a wave traveling with a finite speed (with respect to the adiabatic time), whereas the semiclassical measure of the system will undergo a controllable transformation. If adiabatic time goes to infinity then, by quantum Furstenberg Theorem, the system will become quantum uniquely ergodic. Thus, infinite excitation of a closed system leads to quantum chaos.
... For hyperbolic surfaces, Dyatlov-Jin [DJ18] showed a different kind of restriction on semiclassical measures µ: each such measure should have full support, i.e. µ(U ) > 0 for any nonempty open set U ⊂ S * M . (See also the expository article [Dya17].) ...
... This rules out the fractal counterexamples of the kind found in [AN07a] (which can have entropy close to 1 and thus are not ruled out by the entropy bound (21)). There is no contradiction here since the key new ingredient in [DJ18], the fractal uncertainty principle of Bourgain-Dyatlov [BD18], does not hold for the Walsh quantization used in [AN07a]. On the other hand, linear combinations cµ L +(1−c)δ γ with 0 < c < 1 2 have full support but are ruled out by (21). ...
Preprint
We discuss Shnirelman's Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.
... It will be a key ingredient in the proof of the main theorem of this paper. It has been successfully used to show spectral gaps for convex co-compact hyperbolic surfaces ( [DZ16], [BD17], [DJ18], [DZ18]). A discrete version of the fractal uncertainty principle is also the main ingredient of [DJ17] where the author proved a spectral gap for open quantum maps in a toy model case. ...
... This is not useful for our use (we only needed the first implication) but we found that it could be of independent interest. Our proof is based on the proof of Lemma 5.4 in [DJ18]. We adopt the same notations as in 6.1. ...
Preprint
In this paper, we study the problem of scattering by several strictly convex obstacles, with smooth boundary and satisfying a non eclipse condition. We show, in dimension 2 only, the existence of a spectral gap for the meromorphic continuation of the Laplace operator outside the obstacles. The proof of this result relies on a reduction to an open hyperbolic quantum map, achieved in [arXiv:1105.3128]. In fact, we obtain a spectral gap for this type of objects, which also has applications in potential scattering. The second main ingredient of this article is a fractal uncertainty principle. We adapt the techniques of [arXiv:1906.08923] to apply this fractal uncertainty principle in our context.
... In particular, we will describe full support statements for weak limits -see Theorem 4 and Theorem 8 -proved in [DJ18,DJN21,DJ21]. The key component is the fractal uncertainty principle first introduced by Dyatlov-Zahl [DZ16] and proved by Bourgain-Dyatlov [BD18]. ...
... Another way to characterize how much a measure µ is 'spread out' is by looking at its support, supp µ ⊂ S * M . For surfaces with Anosov geodesic flows, Dyatlov-Jin [DJ18] (in the hyperbolic case) and Dyatlov-Jin-Nonnenmacher [DJN21] (in the general case) showed that the support of every semiclassical measure is the entire S * M : have full support and small entropy: one can for example take a convex combination of the Liouville measure and a measure supported on a closed geodesic. ...
Preprint
We give an overview of the interplay between the behavior of high energy eigenfunctions of the Laplacian on a compact Riemannian manifold and the dynamical properties of the geodesic flow on that manifold. This includes the Quantum Ergodicity theorem, the Quantum Unique Ergodicity conjecture, entropy bounds, and uniform lower bounds on mass of eigenfunctions. The above results belong to the domain of quantum chaos and use microlocal analysis, which is a theory behind the classical/quantum, or particle/wave, correspondence in physics. We also discuss the toy model of quantum cat maps and the challenges it poses for Quantum Unique Ergodicity.
... For negatively curved Riemannian manifolds we believe that it is possible to prove better versions of the BMO estimate and better bounds for the doubling index. We would like to mention an outstanding recent result by Bourgain & Dyatlov [12] and Dyatlov & Jin [32]. knowledge it was first observed by Bers [9]. ...
... In [1], it is proved that on a negatively curved compact Riemannian manifold, every microlocal QL has positive metric entropy and thus cannot be the Dirac along a periodic geodesic (see also [2,11]). It is proved in [10] that, on any compact connected Riemannian surface of constant negative curvature, any microlocal QL has full support, i.e., charges any nonempty open subset of the cosphere bundle. ...
Preprint
We establish some properties of quantum limits on a product manifold, proving for instance that, under appropriate assumptions, the quantum limits on the product of manifolds are absolutely continuous if the quantum limits on each manifolds are absolutely continuous. On a product of Riemannian manifolds satisfying the minimal multiplicity property, we prove that a periodic geodesic can never be charged by a quantum limit.
... This has been shown for examples of arithmetic origin in work of Lindenstrauss [19], [20], and Bourgain-Lindenstrauss [4], Jakobson [17], Holowinsky [14], Holowinsky-Soundararajan [15]. For progress constraining the possible limit measures in general, see Anantharaman [1], Anantharaman-Nonnenmacher [2], Anantharaman-Silberman [3], and Dyatlov-Jin [8]. Even though QUE may fail for certain exceptional sequences of spherical harmonics, VanderKam [28] shows that it does hold with probability tending to 1 for φ λ in a randomly chosen orthonormal basis. ...
Article
We study random spherical harmonics at shrinking scales. We compare the mass assigned to a small spherical cap with its area, and find the smallest possible scale at which, with high probability, the discrepancy between them is small simultaneously at every point on the sphere.
... 4. Other than tori, the only other manifolds for which (1.4) is known for any non-trivial continuous W are compact hyperbolic surfaces. That was proved by Jin [Ji17] using results of Bourgain-Dyatlov [BD16] and Dyatlov-Jin [DJ17]. ...
Article
The purpose of this note is to use the results and methods of our previous work with Bourgain to obtain control and observability by rough functions and sets on rectangular 2-tori. We show that any Lebesgue measurable set of positive measure can be used for observability for the Schroedinger equation. This leads to controllability with rough localization and control functions. For non-empty open sets this follows from the results of Haraux '89 and Jaffard '89 while for square tori and sufficiently long times this can be deduced from the results of Jakobson '97.
... For other applications see [BouDya2,DyaJin,DyaJin2], and for a survey [Dya]. ...
Preprint
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of $\delta$-regular sets in one dimension, but relative to arbitrary axes. Our results remain true if $Y$ is distorted by diffeomorphisms. Our method combines the original approach by Bourgain and Dyatlov, in the more quantitative 2017 rendition by Jin and Zhang, with Cartan set techniques.
... The motivation for the results of Bourgain and Dyatlov [2] is to establish a Fractal Uncertainty Principle for the limit sets of Fuchsian groups. Fractal Uncertainty Principle, as introduced by Dyatlov and Zahl [8], is a powerful harmonic analytic tool used in understanding Pollicott-Ruelle resonances in open dynamical systems [3] and delocalisation of semiclassical limits of eigenfunctions for the Laplacian [7]. ...
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We study when Fourier transforms of Gibbs measures of sufficiently nonlinear expanding Markov maps decay at infinity at a polynomial rate. Assuming finite Lyapunov exponent, we reduce this to a nonlinearity assumption, which we verify for the Gauss map using Diophantine analysis. Our approach uses large deviations and additive combinatorics, which combines the earlier works on the Gibbs measures for Gauss map (Jordan-Sahlsten, 2013) and Fractal Uncertainty Principle (Bourgain-Dyatlov, 2017).
... The Quantum Unique Ergodicity (QUE) conjecture, due to Rudnick and Sarnak [31], predicts that if M has negative curvature, the full sequence in (1.1) should converge. Although fundamental results have been made by Lindenstrauss [27], Anantharaman [5] and Dyatlov-Jin [16] in this direction, the conjecture is still open. In §4.2 we prove: ...
Preprint
We investigate the asymptotic behavior of eigenfunctions of the Laplacian on Riemannian manifolds. We show that Benjamini-Schramm convergence provides a unified language for the level and eigenvalue aspects of the theory. As a result, we present a mathematically precise formulation of Berry's conjecture for a compact negatively curved manifold and formulate a Berry-type conjecture for sequences of locally symmetric spaces. We prove some weak versions of these conjectures. Using ergodic theory, we also analyze the connections of these conjectures to Quantum Unique Ergodicity.
... There are partial results, [2], [7], [47], which hold in great generality on any compact negatively curved manifold that show that concentration on sets of low Haussdorff dimension is not possible (a closed geodesic, for instance). Recently, Dyatlov and Jin [17] have shown that, in the case of surfaces of constant negative curvature, elements in N ( H ) must charge every open set U ⊂ M. ...
Preprint
A semiclassical version of the classical KAM theorem about perturbations of constant vector fields on the torus is obtained. Moreover, given a small and bounded perturbation of a linear Hamiltonian on the torus with constant coefficients, the problem of finding an integrable counterterm that renormalizes the system making it canonically conjugate to the unperturbed one is also addressed in the semiclassical setting. These results are used to obtain a characterization of the sets of semiclassical measures and quantum limits associated to sequences of $L^2$-densities of eigenfunctions for the considered Schr\"odinger operators.
... This has been shown for examples of arithmetic origin in work of Lindenstrauss [22,23], and Bourgain-Lindenstrauss [4], Jakobson [19], Holowinsky [17], and Holowinsky-Soundararajan [16]. For a general metric, work of Anantharaman [1], Anantharaman-Nonnenmacher [2], Anantharaman-Silberman [3], and Dyatlov-Jin [10] places constraints on the measures that arise as quantum limits but it remains unknown whether the uniform measure is the only possibility. ...
Preprint
We prove equidistribution at shrinking scales for the monochromatic ensemble on a compact Riemannian manifold of any dimension. This ensemble on an arbitrary manifold takes a slowly growing spectral window in order to synthesize a random function. With high probability, equidistribution takes place close to the optimal wave scale and simultaneously over the whole manifold. The proof uses Weyl's law to approximate the two-point correlation function of the ensemble, and a Chernoff bound to deduce concentration.
... Theorem 3.3 (valid for any smooth nonnegative q ≡ 0) is the first result of this kind for any manifold. We refer the reader to the introductions to [DJ18b,Ji17a,Ji17b] for an overview of various related results. ...
Preprint
This article provides a broad review of recent developments on the fractal uncertainty principle and their applications to quantum chaos.
... The quantum ergodicity theorem proved by Shnirelman [25,26], Colin de Verdière [9], and Zelditch [29] shows that negative curvature implies convergence along a full subsequence of eigenfunctions, or equivalently on average over the eigenfunctions, but allows many other subsequential limits besides the uniform measure. Although it remains unknown whether the uniform measure is the only possibility, work of Anantharaman [1], Anantharaman-Nonnenmacher [2], Anantharaman-Silberman [7], and Dyatlov-Jin [12] places significant constraints on the measures that arise as quantum limits in general. In examples of arithmetic origin, QUE has been proved in work of Lindenstrauss [20,21], and Bourgain-Lindenstrauss [6], Jakobson [19], Holowinsky [18], Holowinsky-Soundararajan [17]. ...
Preprint
We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of arbitrary dimension, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer's addition formula.
... When considering the Schrödinger equation with potentials, controllability without the GCC were obtained by Burq-Zworski [48], Bourgain-Burq-Zworski [36], AnantharamanMarcì a [18], Bourgain [35], etc., using microlocal techniques. Recently, using results and techniques from Dyatlov-Jin [74], Jin [110] proved the controllability of (1.1.9) on hyperbolic surfaces without the GCC. ...
Thesis
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In this thesis, we study the closely related theories of control, stabilization and propagation of singularities for some linear and nonlinear dispersive partial differential equations. Main results come from the author’s works:[1] Zhu, H., 2016. Stabilization of damped waves on spheres and Zoll surfaces of revolution. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV), to appear.[2] Zhu, H., 2017. Control of three dimensional water waves. arXiv preprint arXiv:1712.06130.[3] Zhu, H., 2018. Propagation of singularities for gravity-capillary water waves. arXiv preprint arXiv:1810.09339.In [1] we studied the stabilization of the damped wave equation on Zoll surfaces of revolution. We gave an example where the region of damping is at the borderline of the geometric control condition, yet the damped waves exhibit a uniform exponential decay of energy, generalizing an example of Lebeau.In [2] we studied the controllability of the gravity-capillary water wave equation. Under the geometric control condition, we proved in arbitrary spatial dimension the exact controllability for spatially periodic small data. This generalizes a result of Alazard, Baldi and Han-Kwan for the 2D gravity-capillary water wave equation.In [3] we studied the propagation of singularities for the gravity-capillary water wave equation. We defined the quasi-homogeneous wavefront set, generalizing the wavefront set of H¨ ormander and the homogeneous wavefront set of Nakamura, and proved propagation results for quasi-homogeneous wavefront sets by the gravity-capillary water wave equation. As corollaries, we obtained local and microlocal smoothing effects for gravity-capillary water waves with sufficient spatial decay.
... Logarithmic decay from the damping within arbitrarily small open sets has been well investigated in [Leb93,LR97,Bur98]. Note due to recent Schrödinger observability results [DJ18,Jin20], interior damping on any open set of a compact hyperbolic surface also gives exponential decay. ...
Preprint
We study the decay of global energy for the wave equation with H\"older continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.
... Over the years, a particular attention has been drawn towards Riemannian manifolds whose geodesic flow is ergodic since in this case, up to extraction of a density-one subsequence, the set of Quantum Limits is reduced to the Liouville measure, a phenomenon which is called Quantum Ergodicity (see for example [Shn74], [Col85], [Zel87]). More recently, the results [Ana08] and [DJ18] gave more precise results in the negative curvature case, using in the first case the notion of metric entropy and in the second one the fractal uncertainty principle. For compact arithmetic surfaces, a detailed study of invariant measures lead to the resolution of the Quantum Unique Ergodicity conjecture for these manifolds, meaning that the extraction of a density-one subsequence in the Quantum Ergodicity result is even not necessary for these particular manifolds ( [Lin06]). ...
Thesis
In this thesis at the boundary between analysis and geometry, we study some subelliptic partial differential equations (PDEs) with modern tools coming from sub-Riemannian geometry and microlocal analysis. We first study the controllability and observability of some subelliptic PDEs: we show that in directions requiring more brackets to be generated, the propagation of energy (and hence the observability) takes more time. Our results apply with full generality to linear subelliptic wave equations, but also to some Schrödinger-type and damped wave equations. Then, we study the propagation of singularities in subelliptic wave equations: we show that singularities propagate only along null-bicharacteristics and abnormal extremal lifts of singular curves. This result makes a bridge with classical notions in sub-Riemannian geometry. We illustrate it in the Martinet case: we construct initial data whose singularities propagate along any singular curve at any speed between 0 and 1. Finally, we study the eigenfunctions of some families of subelliptic Laplacians, in the high-frequency limit: we show that their limits, called quantum limits, can be decomposed in an infinite number of pieces, corresponding to an infinite number of dynamics on the underlying manifold.
... For negatively curved Riemannian manifolds we believe that it is possible to prove better versions of the BMO estimate and better bounds for the doubling index. We would like to mention an outstanding recent result by Bourgain & Dyatlov [12] and Dyatlov & Jin [32]. knowledge it was first observed by Bers [9]. ...
Preprint
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This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.
... To keep track of powers of h in the remainders, we use a special class of symbols than one used in [DJ18]. ...
Preprint
The point of this paper is to improve the reverse Agmon estimate discussed in \cite{TW} with assuming that the Schrodinger operator $P(h) = - h^2 \Delta_g + V - E(h)$, $E(h)\to E$ as $h\to 0^+$, is analytic on a compact, real-analytic Riemannian manifold $(M,g)$. In this paper, by considering a Neumann problem with applying Poisson representation and exterior mass estimates on hypersurfaces, we can prove an improved reverse Agmon estimate on a hypersurface.
... On the other side of the dynamics lie manifolds of negative curvature; characterizing the set of Quantum Limits in this case is part of the Quantum Unique Ergodicity conjecture [28]: the conjecture implies that the only quantum limit is the Riemannian volume. The literature is vast in this setting; see, among many others, [1,4,8,9,14,19,23,29,34]. ...
Preprint
We characterize quantum limits and semi-classical measures corresponding to sequences of eigenfunctions for systems of coupled quantum harmonic oscillators with arbitrary frequencies. The structure of the set of semi-classical measures turns out to depend strongly on the arithmetic relations between frequencies of each decoupled oscillator. In particular, we show that as soon as these frequencies are not rational multiples of a fixed fundamental frequency, the set of semi-classical measures is not convex and therefore, infinitely many measures that are invariant under the classical harmonic oscillator are not semi-classical measures.
... The motivation for the results of Bourgain and Dyatlov [3] is to establish a Fractal Uncertainty Principle for the limit sets of Fuchsian groups. Fractal Uncertainty Principles, which have a few different forms, were introduced by Dyatlov and Zahl [14] as a powerful harmonic analytic tool to understand for example Pollicott-Ruelle resonances in open dynamical systems [4], and delocalisation of semiclassical limits of eigenfunctions for the Laplacian in Quantum Chaos [13]. We refer to the survey of Dyatlov [15] on more about the history and ideas behind Fractal Uncertainty Principles. ...
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FOR THE LATEST FILE VERSION PLEASE SEE ARXIV. https://arxiv.org/abs/2009.01703 We study the Fourier transforms $\widehat{\mu}(\xi)$ of Gibbs measures $\mu$ for uniformly expanding maps $T$ of bounded distortions on Cantor sets with strong separation condition. When $T$ is totally non-linear and Hausdorff dimension of $\mu$ is large enough, then $\widehat{\mu}(\xi)$ decays at a polynomial rate as $|\xi| \to \infty$. Edit: we believe that the dimension assumption can be removed. Version 2: corrections made to main assumptions. Ammended some typos, and expanded the introduciton.
... There are remarkable applications of microlocal analysis and related ideas in many fields of mathematics. Classical examples include spectral theory and the Atiyah-Singer index theorem, and more recent examples include scattering theory [4], behavior of chaotic systems [5], general relativity [6], and inverse problems. ...
Article
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This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019.
... A recent breakthrough on the QUE problem for hyperbolic surfaces is the following result due to Dyatlov-Jin [DJ18]. We use semiclassical notation where ℏ = −1 and denote eigenfunctions by ℏ or more simply ℏ as in (10). ...
... Despite counterexamples demonstrating that ergodicity alone is insufficient for quantum unique ergodicity, there is numerical evidence to support the conjecture in the presence of negative curvature [AS93,HR92]. In addition there are striking results of Anantharaman and Nonnenmacher [AN07,Ana08] and Dyatlov and Jin [DJ18] regarding the entropy and support of possible limits of quantum probability measures. Moreover, Lindenstrauss [Lin06] (with an extension by Soundararajan [Sou10] for the non-compact case), proved that the quantum unique ergodicity conjecture holds for Hecke-Laplace eigenfunctions on arithmetic surfaces. ...
Preprint
A finite group $G$ is called $C$-quasirandom (by Gowers) if all non-trivial irreducible complex representations of $G$ have dimension at least $C$. For any unit $\ell^{2}$ function on a finite group we associate the quantum probability measure on the group given by the absolute value squared of the function. We show that if a group is highly quasirandom, in the above sense, then any Cayley graph of this group has an orthonormal eigenbasis of the adjacency operator such that the quantum probability measures of the eigenfunctions put close to the correct proportion of their mass on subsets of the group that are not too small.
Preprint
We study dynamical properties of the billiard flow on $3$ dimensional convex polyhedrons away from "pockets" and establish a finite tube condition for rational polyhedrons that extends well-known results in dimension $2$. Furthermore, we establish a new quantitative estimate for lengths of periodic tubes in irrational polyhedrons. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition that extends previously known estimates for periodic boundary conditions, which we believe should be of independent interest.
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We derive a central limit theorem for the mean-square of random waves in the high-frequency limit over shrinking sets. Our proof applies to any compact Riemannian manifold of dimension 3 or higher, thanks to the universality of the local Weyl law. The key technical step is an estimate capturing some cancellation in a triple integral of Bessel functions, which we achieve using Gegenbauer’s addition formula.
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Fractal uncertainty principle states that no function can be localized in both position and frequency near a fractal set. This article provides a review of recent developments on the fractal uncertainty principle and of their applications to quantum chaos, including lower bounds on mass of eigenfunctions on negatively curved surfaces and spectral gaps on convex cocompact hyperbolic surfaces.
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We prove a uniform lower bound on Cauchy data on an arbitrary curve on a negatively curved surface using the Dyatlov-Jin(-Nonnenmacher) observability estimate on the global surface. In the process, we prove some further results about defect measures of restrictions of eigenfunctions to a hypersurface.
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We show that any nonempty open set on a hyperbolic surface provides observability and control for the time dependent Schrödinger equation. The only other manifolds for which this was previously known are flat tori [11–13]. The proof is based on the main estimate in [10] and standard arguments in control theory. © 2018 International Press of Boston, Inc.. All rights reserved.
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We prove equidistribution at shrinking scales for the monochromatic ensemble on a compact Riemannian manifold of any dimension. This ensemble on an arbitrary manifold takes a slowly growing spectral window in order to synthesize a random function. With high probability, equidistribution takes place close to the optimal wave scale and simultaneously over the whole manifold. The proof uses Weyl’s law to approximate the two-point correlation function of the ensemble, and a Chernoff bound to deduce concentration.
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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 Schrödinger 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 (see Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339] and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794]), which considered the setting of surfaces of constant negative curvature. The proofs use the strategy of Semyon Dyatlov and Long Jin [Acta Math. 220 (2018), pp. 297–339 and Long Jin [Comm. Math. Phys. 373 (2020), pp. 771–794] and rely on the fractal uncertainty principle of Jean Bourgain and Semyon Dyatlov [Ann. of Math. (2) 187 (2018), pp. 825–867]. 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 by Semyon Dyatlov and Joshua Zahl [Geom. Funct. Anal. 26 (2016), pp. 1011–1094]. Instead, our argument uses Egorov’s theorem up to local Ehrenfest time and the hyperbolic parametrix of Stéphane Nonnenmacher and Maciej Zworski [Acta Math. 203 (2009), pp. 149–233], together with the C 1 + C^{1+} regularity of the stable/unstable foliations.
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We discuss Shnirelman’s Quantum Ergodicity Theorem, giving an outline of a proof and an overview of some of the recent developments in mathematical Quantum Chaos.
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We describe the complex poles of the power spectrum of correlations for the geodesic flow on compact hyperbolic manifolds in terms of eigenvalues of the Laplacian acting on certain natural tensor bundles. These poles are a special case of Pollicott-Ruelle resonances, which can be defined for general Anosov flows. In our case, resonances are stratified into bands by decay rates. The proof also gives an explicit relation between resonant states and eigenstates of the Laplacian.
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Given any compact hyperbolic surface $M$, and a closed geodesic on $M$, we construct of a sequence of quasimodes on $M$ whose microlocal lifts concentrate positive mass on the geodesic. Thus, the Quantum Unique Ergodicity (QUE) property does not hold for these quasimodes. This is analogous to a construction of Faure-Nonnenmacher-De Bi\`evre in the context of quantized cat maps, and lends credence to the suggestion that large multiplicities play a role in the known failure of QUE for certain "toy models" of quantum chaos. We moreover conjecture a precise threshold for the order of quasimodes needed for QUE to hold--- the result of the present paper shows that this conjecture, if true, is sharp.
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This is a survey of recent results on quantum ergodicity, specifically on the large energy limits of matrix elements relative to eigenfunctions of the Laplacian. It is mainly devoted to QUE (quantum unique ergodicity) results, i.e. results on the possible existence of a sparse subsequence of eigenfunctions with anomalous concentration. We cover the lower bounds on entropies of quantum limit measures due to Anantharaman, Nonnenmacher, and Rivi\`ere on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE result of Hassell in the case of the Bunimovich stadium. We include some discussion of Hecke eigenfunctions and recent results of Soundararajan completing Lindenstrauss' QUE result, in the context of matrix elements for Fourier integral operators. Finally, in answer to the potential question `why study matrix elements' it presents an application of the author to the geometry of nodal sets.
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In this paper we study some problems arising from the theory of Quantum Chaos, in the context of arithmetic hyperbolic manifolds. We show that there is no strong localization (“scarring”) onto totally geodesic submanifolds. Arithmetic examples are given, which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom.
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. In this paper we prove microlocal version of the equidistribution theorem for Wigner distributions associated to Eisenstein series on PSL 2 (Z)nPSL 2 (R). This generalizes a recent result of W. Luo and P. Sarnak who prove equidistribution for PSL 2 (Z)nH. The averaged versions of these results have been proven by Zelditch for an arbitrary finite-volume surface, but our proof depends essentially on the presence of Hecke operators and works only for congruence subgroups of SL 2 (R). In the proof the key estimates come from applying Meurman's and Good's results on Lfunctions associated to holomorphic and Maass cusp forms. One also has to use classical transformation formulas for generalized hypergeometric functions of a unit argument. R' esum' e. Nous donnons la preuve d'une version microlocale d'un resultat de W. Luo et P. Sarnak concernant la r'epartition asymptotique des fonctions de Wigner associ'ees aux series d'Eisenstein sur PSL 2 (Z)nPSL 2 (R). La preuve utilise les op'erateur...
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Let N be a compact hyperbolic manifold, an embedded totally geodesic submanifold, and let be the semiclassical Laplace–Beltrami operator. For any we explicitly construct families of quasimodes of energy width at most which exhibit a 'strong scar' on M in that their microlocal lifts converge weakly to a probability measure which places positive weight on . An immediate corollary is that any invariant measure on occurs in the ergodic decomposition of the semiclassical limit of certain quasimodes of width .
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Introduction 1. Local symplectic geometry 2. The WKB-method 3. The WKB-method for a potential minimum 4. Self-adjoint operators 5. The method of stationary phase 6. Tunnel effect and interaction matrix 7. h-pseudodifferential operators 8. Functional calculus for pseudodifferential operators 9. Trace class operators and applications of the functional calculus 10. More precise spectral asymptotics for non-critical Hamiltonians 11. Improvement when the periodic trajectories form a set of measure 0 12. A more general study of the trace 13. Spectral theory for perturbed periodic problems 14. Normal forms for some scalar pseudodifferential operators 15. Spectrum of operators with periodic bicharacteristics References Index Index of notation.
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We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a compact manifold of variable negative curvature - or more generally, assuming only that the geodesic flow has the Anosov property. We prove that the Wigner measures associated to eigenfunctions cannot concen- trate entirely on sets of small topological entropy under the action of the geodesic flow, such as, for instance, closed geodesics. 1. Introduction, statement of results We consider a compact Riemannian manifold M of dimension d ‚ 2, and as- sume that the geodesic flow (gt)t2R, acting on the unit tangent bundle of M, has a "chaotic" behaviour; this refers to certain asymptotic properties of the flow when time t tends to infinity: ergodicity, mixing, hyperbolicity... Here we mean that the geodesic flow has the Anosov property. The name "quantum chaos" expresses the belief that the chaotic properties of the flow should still be visible in the correspond- ing quantized dynamical system: that is, according to the Schrodinger equation, the unitary flow ¡ exp(i~t¢ 2 ) ¢ t2R acting on the Hilbert space L2(M) - where ¢ stands for the Laplacian on M and~ is something proportional to the Planck constant. At the quantum level, one expects that the chaotic features should express themselves in certain behaviours of the eigenfunctions of the Laplacian, or in the distribution of its eigenvalues (see (Sa95)). These ideas rely on the fact that the quantum flow ¡ exp(i~t¢ 2 ) ¢ t2R converges, in a sense to be precised below, to the classical flow (g t) in the so-called "semi-classical limit" ~ ¡! 0: one likes to imagine that "for ~ small" the qualitative behaviour of quantum system will be related to that of the classical flow. The convergence of the quantum flow to the classical flow is stated precisely in the Egorov theorem. Let us consider one of the usual quantization procedures, say Op~, which associates an operator Op~(a) acting on L2(M) to every smooth compactly supported function a 2 C1
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We classify measures on the locally homogeneous space \SL(2,R) ◊ L which are invariant and have positive entropy un- der the diagonal subgroup of SL(2,R) and recurrent under L. This classification can be used to show arithmetic quantum unique er- godicity for compact arithmetic surfaces, and a similar but slightly weaker result for the finite volume case. Other applications are also presented. In the appendix, joint with D. Rudolph, we present a maximal ergodic theorem, related to a theorem of Hurewicz, which is used in the proof of the main result.
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We report on some recent striking advances on the quantum unique ergodicity or "QUE" conjecture, concerning the distribution of large frequency eigenfunctions of the Laplacian on a negatively curved manifold. The account falls naturally into two cate-gories. The first concerns the general conjecture where the tools are more or less limited to micro-local analysis and the dynamics of the geodesic flow. The second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue. The recent account by Zelditch [Zel4] covers some of the same material and his and our discussion complement each other nicely, as he goes into more detail with the first category and we with the second. This note is not meant to be a survey of these topics and the discussion is not chrono-logical . Our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context.
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In this paper I give simple proofs of Raghunathan’s conjectures for SL(2,R). These proofs incorporate in a simplified form some of the ideas and methods I used to prove the Raghunathan’s conjectures for general connected Lie groups.
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H. Furstenberg showed that horocycle flows on compact manifolds of constant negative curvature are uniquely ergodic. This paper generalizes his result to the case of variable negative curvature, in the more general context of flows whose orbits are the unstable manifolds of certain Anosov flows.
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We study the high-energy asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface $M$ of Anosov type. To do this, we look at families of distributions associated to them on the cotangent bundle $T^*M$ and we derive entropic properties on their accumulation points in the high-energy limit (the so-called semiclassical measures). We show that the Kolmogorov-Sinai entropy of a semiclassical measure $\mu$ for the geodesic flow $g^t$ is bounded from below by half of the Ruelle upper bound; that is, \[h_{KS}(\mu,g)\geq \frac{1}{2}\int_{S^*M} \chi^+(\rho)\ d\!\mu(\rho),\] where $\chi^+(\rho)$ is the upper Lyapunov exponent at point $\rho$ .
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We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite $n\times n$ matrices (any quotient for $n=3$, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Lebesgue component. This is consistent with the conjecture that the limit measures are absolutely continuous.
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We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of nonpositive sectional curvature. To do this, we look at sequences of distributions associated to them and we study the entropic properties of their accumulation points, the so-called semiclassical measures. Precisely, we show that the Kolmogorov–Sinai entropy of a semiclassical measure μ for the geodesic flow g t is bounded from below by half of the Ruelle upper bound, i.e. $$h_{KS}(\mu,g)\geq \frac{1}{2} \int\limits_{S^*M} \chi^+(\rho) {\rm d} \mu(\rho),$$where χ +(ρ) is the upper Lyapunov exponent at point ρ. The main strategy is the same as in Rivière (Duke Math J, arXiv:0809.0230, 2008) except that we have to deal with weakly chaotic behavior.
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Résumé Nous donnons la preuve d'une généralisation d'un résultat récent de S. Zelditch concernant la répartition asymptotique des fonctions propres du laplacien sur une variété compacte dont le flot géodésique est ergodique.
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We eliminate the possibility of "escape of mass" for Hecke-Maass forms of large eigenvalue for the modular group. Combined with the work of Lindenstrauss, this establishes the Quantum Unique Ergodicity conjecture of Rudnick and Sarnak for the modular surface SL_2(Z)\H.
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We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized.
  • Maciej Zworski
Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, AMS, 2012.
Strong scarring of logarithmic quasimodes
  • Suresh Eswarathasan
  • Stéphane Nonnenmacher
Suresh Eswarathasan and Stéphane Nonnenmacher, Strong scarring of logarithmic quasimodes, to appear in Ann. Inst. Fourier, arXiv:1507.08371.