Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.
We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on Riemannian surfaces with finitely many hyperbolic cusps and ergodic geodesic flow.
Suppose that Omega is a bounded, piecewise smooth Euclidean domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Omega with various boundary conditions are quantum ergodic if the classical billiard map (on the ball bundle of the boundary of Omega) is ergodic. Our proof is based on the classical observation that the boundary value of an interior eigenfunction is an eigenfunction of an operator F_h on the boundary of Omega, where h^{-2} is equal to the eigenvalue. In the case of the Neumann boundary condition, F_h is the boundary integral operator induced…
We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of…
Consider a point scatterer (the Laplacian perturbed by a delta-potential) on the standard three-dimensional flat torus. Together with the eigenfunctions of the Laplacian which vanish at the point, this operator has a set of new, perturbed eigenfunctions. In a recent paper, the author was able to show that all of the perturbed eigenfunctions are uniformly distributed in configuration space. In this paper we prove that almost all of these eigenfunctions are uniformly distributed in phase space, i.e. we prove quantum ergodicity for the subspace of the perturbed eigenfunctions. An analogue result …
The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(λ) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(λ)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivière small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, w…
For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.
We prove an analogue of Shnirelman, Zelditch and Colin de Verdiere's Quantum Ergodicity Theorems in a case where there is no underlying classical ergodicity. The system we consider is the Laplacian with a delta potential on the square torus. There are two types of wave functions: old eigenfunctions of the Laplacian, which are not affected by the scatterer, and new eigenfunctions which have a logarithmic singularity at the position of the scatterer. We prove that a full density subsequence of the new eigenfunctions equidistribute in phase space. Our estimates are uniform with respect to the cou…
We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is …
We study the semiclassical behaviour of eigenfunctions of quantum systems with ergodic classical limit. By the quantum ergodicity theorem almost all of these eigenfunctions become equidistributed in a weak sense. We give a simple derivation of an upper bound of order $\abs{\ln\hbar}^{-1}$ on the rate of quantum ergodicity if the classical system is ergodic with a certain rate. In addition we obtain a similar bound on transition amplitudes if the classical system is weak mixing. Both results generalise previous ones by Zelditch. We then extend the results to some classes of quantised maps on th…
Consider the eigenfunctions $u$ for a ree rectangular membrane wo that $-Δu=λu$ on $\mathcal R(c,d)=(0,d)\times(0,d)$. In this note we show that if if $u>0$ on $\partial \mathcal R(c,d)) then $u\equiv C$ for some positive constant.
This is a report for the 2003 Forges Les Eaux PDE conference on recent results with A. Hassell on quantum ergodicity of boundary traces of eigenfunctions on domains with ergodic billiards, and of work in progress with Hassell and Sogge on norms of boundary traces. Related work by Burq, Grieser and Smith-Sogge is also discussed.
In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.
Let $(Ω,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial Ω.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions decay exponentially into the interior in terms of the Dirichlet-to-Neumann eigenvalues and give a sharp rate of decay to first order at the boundary. The proof uses the Poisson representation for the Steklov eigenfunctions combined with sharp $h$-microlocal concentration estimates for the boundary Dirichlet-to-Neumann eigenfunctions near the cosph…
In this paper we consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions. The main result is a quantitative estimate on the $L^2$ mass of eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face. In particular, eigenfunctions cannot concentrate completely at any one single point. The technique of proof is to use the commutator ideas from the recent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.
We prove a quantum ergodic restriction theorem for the Cauchy data of a sequence of quantum ergodic eigenfunctions on a hypersurface $H$ of a Riemannian manifold $(M, g)$. The technique of proof is to use a Rellich type identity to relate quantum ergodicity of Cauchy data on $H$ to quantum ergodicity of eigenfunctions on the global manifold $M$. This has the interesting consequence that if the eigenfunctions are quantum unique ergodic on the global manifold $M$, then the Cauchy data is automatically quantum unique ergodic on $H$ with respect to operators whose symbols vanish to order one on th…
The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunction on the boundary of a smooth boundary domain $\mathbb R^n$. The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Lin and Bellova \cite{BL}. Furthermore, we show that the vanishing order of Steklov eigenfunction is everywhere less than $Cλ$ where $λ$ is the Steklov eigenvalue and $C$ depends only on $Ω$.
Let $e_ł(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $Δ_M$ on a compact Riemannian manifold $M$ without boundary: $Δ_M e_ł=ł^2 e_ł$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, where $C$ is a positive constant depending only on $M$.
Let $u$ be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let $-λ^2$ be the corresponding eigenvalue. We consider the problem of estimating the maximum of $u$ in terms of $λ$, for large $λ$, assuming $u$ is $L^2$-normalized. We prove that $\max_M u\leq C_M λ^{(n-1)/2}$, which is optimal for some $M$. Our proof simplifies some of the arguments used before for such problems. In order to make the article accessible to non-specialists, we review the 'wave equation method' (which has become standard in asymptotic eigenvalue …
We prove sharp upper and lower bounds for the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary. The argument involves frequency function methods for harmonic functions in the interior of the surface as well as the construction of exponentially accurate approximations for the Steklov eigenfunctions near the boundary.
Let $e_ł(x)$ be a Neumann eigenfunction with respect to the positive Laplacian $Δ$ on a compact Riemannian manifold $M$ with boundary such that $Δ\, e_ł=ł^2 e_ł$ in the interior of $M$ and the normal derivative of $e_ł$ vanishes on the boundary of $M$. Let $χ_λ$ be the unit band spectral projection operator associated with the Neumann Laplacian and $f$ a square integrable function on $M$. We show the following gradient estimate for $χ_λ\,f$ as $λ\geq 1$: $\|\nabla\ χ_ł f\|_\infty\leq Cł\|χ_ł\f\|_\infty+ł^{-1}\|Δ χ_ł f\|_\infty$, where $C$ is a positive constant depending only on $M$. As a coro…
It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any …
Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for…
This paper is part of a series concerning the isospectral problem for an ellipse. In this paper, we study Cauchy data of eigenfunctions of the ellipse with Dirichlet or Neumann boundary conditions. Using many classical results on ellipse eigenfunctions, we determine the microlocal defect measures of the Cauchy data of the eigenfunctions. The ellipse has integrable billiards, i.e. the boundary phase space is foliated by invariant curves of the billiard map. We prove that, for any invariant curve $C$, there exists a sequence of eigenfunctions whose Cauchy data concentrates on $C$. We use this re…
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 Rivière on compact Riemannian manifolds with Anosov flow. These lower bounds give new constraints on the possible quantum limits. We also cover the non-QUE re…
In this thesis, we investigate quantum ergodicity for two classes of Hamiltonian systems satisfying intermediate dynamical hypotheses between the well understood extremes of ergodic flow and quantum completely integrable flow. These two classes are mixed Hamiltonian systems and KAM Hamiltonian systems. Hamiltonian systems with mixed phase space decompose into finitely many invariant subsets, only some of which are of ergodic character. It has been conjectured by Percival that the eigenfunctions of the quantisation of this system decompose into associated families of analogous character. The fi…
This paper is a proceedings version of \cite{CHT-I}, in which we state a Quantum Ergodicity (QE) theorem on a 3D contact manifold, and in which we establish some properties of the Quantum Limits (QL). We consider a sub-Riemannian (sR) metric on a compact 3D manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We state a QE theorem for the eigenfunctions of any associated sR Laplacian, under the assumption that the Reeb flow is ergodic. The limi…
This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L^p norms of the restrictions of eigenfunctions to broken geodesics.
We prove that the Cauchy data of Dirichlet or Neumann $Δ$- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e. a point at which a positive measure of geodesics leaving the point return to the point. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds.
We give a new and self-contained proof of a generic version of the (former) Helgason conjecture. It says that for generic spectral parameters the Poisson transform is a topological isomorphism, with the inverse given by a boundary value map. Following Oshima's approach to a simplified definition of boundary values, and using the earlier work of Baouendi and Goulaouic on Fuchsian type equations, our proof is along the lines of our earlier work in the rank one distributional case, and works for both the hyperfunction and the distribution setting.
We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin domain. The boundary of the domain is assumed to be locally periodic. When the thickness of the domain $\varepsilon$ tends to zero, the eigenvalues are of order $\varepsilon^{-2}$ and described in terms of the first eigenvalue $μ(x_1)$ of an auxiliary spectral cell problem parametrized by $x_1$, while the eigenfunctions localize with rate $\sqrt{\varepsilon}$.
We present a comprehensive study of new discoveries on the spectral patterns of elastic transmission eigenfunctions, including boundary localisation, surface resonance, and stress concentration. In the case where the domain is radial and the underlying parameters are constant, we give rigorous justifications and derive a thorough understanding of those intriguing geometric and physical patterns. We also present numerical examples to verify that the same results hold in general geometric and parameter setups.
We investigate the measure of nodal sets for Robin and Neumann eigenfunctions in the domain and on the boundary of the domain. A polynomial upper bound for the interior nodal sets is obtained for Robin eigenfunctions in the smooth domain. For the analytic domain, the sharp upper bounds of the interior nodal sets was shown for Robin eigenfunctions. More importantly, we obtain the sharp upper bounds for the boundary nodal sets of Neumann eigenfunctions with new quantitative global Carleman estimates. Furthermore, the sharp doubling inequality and vanishing order of Robin eigenfunctions on the bo…
We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.
We show that Steklov eigenfunctions in a bounded Lipschitz domain have wavelength dense nodal sets near the boundary, in contrast to what can happen deep inside the domain. As a converse, in a two-dimensional Lipschitz domain $Ω$, we prove that any nodal domain of a Steklov eigenfunction contains a half-ball centered at $\partialΩ$ of radius $c_Ω/λ$.
We study a coupled deterministic system of vorticity evolution and salinity transport equations, with spatially correlated white noise on the boundary. This system may be considered as a model for gravity currents in oceanic fluids. The noise is due to uncertainty in salinity flux on fluid boundary. After transforming this system into a random dynamical system, we first obtain an asymptotic estimate of enstrophy evolution, and then show that the system is ergodic under suitable conditions on mean salinity input flux on the boundary, Prandtl number and covariance of the noise.
Let $Ω$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_λ$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_λ$ does not exceed $C(Ω)\sqrtλ$. This result is new even for the case of domains with $C^\infty$-smooth boundary.
This paper is devoted to the analysis of Steklov eigenvalues and Steklov eigenfunctions on a class of warped product Riemannian manifolds $(M,g)$ whose boundary $\partial M$ consists in two distinct connected components $Γ_0$ and $Γ_1$. First, we show that the Steklov eigenvalues can be divided into two families $(λ_m^\pm)_{m \geq 0}$ which satisfy accurate asymptotics as $m \to \infty$. Second, we consider the associated Steklov eigenfunctions which are the harmonic extensions of the boundary Dirichlet to Neumann eigenfunctions. In the case of symmetric warped product, we prove that the Stekl…
We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the fre…
This paper is concerned with the intrinsic geometric structures of conductive transmission eigenfunctions. The geometric properties of interior transmission eigenfunctions were first studied in [9]. It is shown in two scenarios that the interior transmission eigenfunction must be locally vanishing near a corner of the domain with an interior angle less than $π$. We significantly extend and generalize those results in several aspects. First, we consider the conductive transmission eigenfunctions which include the interior transmission eigenfunctions as a special case. The geometric structures e…
In this article, we study, as the coefficient $s\to\infty$, the asymptotic behavior of the principal eigenvalue of $$-φ''(x)-2sm'(x)φ'(x)+c(x)φ(x)=λ_sφ(x),\ \ 0<x<1,$$ supplemented by different boundary conditions. This problem is relevant to nonlinear propagation phenomena in reaction-diffusion equations. The main point is that the advection (or drift) term $m$ allows natural degeneracy. For instance, $m$ could be constant on $[a,b]\subset[0,1]$. Depending on the behavior of $m$ near the neighbourhood of the end points $a,\,b$, the limiting value could be the principal eigenvalue of $$-φ''(x)…
We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter $1/\varepsilon$ in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the $j$th eigenfunction can be approximated by a…
It has been empirically observed that eigenfunctions of Laplace's equation $-Δϕ= λϕ$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has some nontrivial implications in acoustics that has been put to real-life use (sound attenuation by noise-protective walls); this short paper describes the mathematical mechanism responsible for this and describes the quantitative strength of the phenomenon for some examples.
The purpose of this paper is to give a simple proof of sharp $L^\infty$ estimates for the eigenfunctions of the Dirichlet Laplacian on smooth compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ with boundary $\partial M$ and then to use these estimates to prove new estimates for Bochner Riesz means in this setting. Our eigenfunction estimates involve estimating the $L^2\to L^\infty$ mapping properties of the operators $χ_λ$ which project onto unit bands of the spectrum of the square root of the Laplacian. These generalize the recent estimates of D. Grieser for individual eigenfunctions.…
We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called ``pockets''. We prove there are only finitely many immersed periodic tubes missing the pockets and moreover establish a new quantitative estimate for the lengths of such tubes. This extends well-known results in dimension $2$. 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 on irrational polyh…
We consider the Burgers equation posed on the outer communication region of a Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid flow under consideration, we study the propagation and interaction of shock waves under the effect of random forcing. First of all, considering the initial and boundary value problem with boundary data prescribed in the vicinity of the horizon, we establish a generalization of the Hopf--Lax--Oleinik formula, which takes the curved geometry into account and allows us to establish the existence of bounded variation solutions. To this end, we …
We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori $\mathbb{R}^d/2 π\mathbb{Z}^d$ in dimensions $d=2,3$. Despite quantum ergodicity holding for the set of "new" eigenfunctions we show that there is scarring in the momentum representation for $d=2,3$, as well as in the position representation for $d=2$ (i.e., the eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues.) For $d=3$, scarred eigenstates are quite rare, but for $d=2$ scarring in the momentum representation is very common --- with $N_{2}(x)…
Let $e_ł(x)$ be an eigenfunction with respect to the Dirichlet Laplacian $Δ_N$ on a compact Riemannian manifold $N$ with boundary: $Δ_N e_ł=ł^2 e_ł$ in the interior of $N$ and $e_ł=0$ on the boundary of $N$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, where $C$ is a positive constant depending only on $N$. In the proof, we use a basic geometrical property of nodal sets of eigenfunctions and elliptic apriori estimates.
Consider the transmission eigenvalue problem for $u \in H^1(Ω)$ and $v\in H^1(Ω)$ associated with $(Ω; σ, \mathbf{n}^2)$, where $Ω$ is a ball in $\mathbb{R}^N$, $N=2,3$. If $σ$ and $\mathbf{n}$ are both radially symmetric, namely they are functions of the radial parameter $r$ only, we show that there exists a sequence of transmission eigenfunctions $\{u_m, v_m\}_{m\in\mathbb{N}}$ associated with $k_m\rightarrow+\infty$ as $m\rightarrow+\infty$ such that the $L^2$-energies of $v_m$'s are concentrated around $\partialΩ$. If $σ$ and $\mathbf{n}$ are both constant, we show the existence of transmi…
