Sinan Ariturk
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.
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In this paper we will prove the nodal line $N$ of the second eigenfunction of the Laplacian over some simply connected concave domain $Ω$ in $\mathbb{R}^2$ must intersect the boundary $\partialΩ$ at exactly two points.
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.
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 …
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. R{é}sum{é}.
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 study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superlevel sets of the eigenfunction stabilise as they approach the maximum, uniformly with respect to the eccentricity of the domain itself. This is achieved by obtaining quantitatively sharp second derivative estimates, which are consistent with the shape of the superlevel sets. In particular, we prove that the eigenfunction is concave (rather than merely log concave) in an entire superlevel set near its maximum. By esti…
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…
We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for $L^1$-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.
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.
A Priori Log-Concavity Estimates for Dirichlet Eigenfunctions
针对Riemann流形凸区域上的第一Dirichlet特征函数，研究在其对数凹假设下推导$\log u$的Hessian定量先验估计。
On a convex bounded Euclidean domain, the ground state for the Laplacian with Neumann boundary conditions is a constant, while the Dirichlet ground state is log-concave. The Robin eigenvalue problem can be considered as interpolating between the Dirichlet and Neumann cases, so it seems natural that the Robin ground state should have similar concavity properties. In this paper we show that this is false, by analysing the perturbation problem from the Neumann case. In particular we prove that on polyhedral convex domains, except in very special cases (which we completely classify) the variation …
The strong log-concavity for first eigenfunction of the Ornstein-Uhlenbeck operator in the class of convex bodies
本文证明了有界凸域上 Ornstein-Uhlenbeck 算子的第一 Dirichlet 特征值具有强对数凹性，改进了已有结论，并刻画了凸体类中主频率 Brunn-Minkowski 不等式的等号情形。
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures corresponding to eigenfunctions. These measures describe simultaneously the concentration and oscillation effects developed by a sequence of eigenfunctions. We present some results showing how to obtain invariant semiclassical measures from eigenfunctions with prescribed symmetries. As an ap…
Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$…
We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hdσ_H\Big| $$ where $H\subset M$ is a submanifold and $σ_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\leq k\leq n$ and in particular, gives a new approach to understanding $\|u_h\|_{L^\infty(M)}$. The method, developed in the author's recent work together with Y. Canzani and J. Toth, relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By…
- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori.
Concentration of eigenfunctions on singular Riemannian manifolds
本文研究边界度量奇异的紧黎曼流形，将 Laplace–Beltrami 算子表示为 Grushin 算子加奇异势，并确定超临界 regime 下高频特征函数在边界聚集的速率与轮廓，应用于气态行星声学模式。
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…
To $1/2$-logconcavity and beyond: Geometric properties of Dirichlet eigenfunctions
本文证明在有界开凸域上，Laplacian 或 Ornstein-Uhlenbeck 算子的第一 Dirichlet 本征函数对于所有 α∈(0,1/2] 是 α-对数凹的，并给出了缩放参数 κ 的显式阈值，同时建立了局部凸性结果，并提供了反例说明无缩放的 1/2-对数凹性在 Schrödinger 算子和加权 Laplacian 情形下可能不成立。
We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $Γ$ be the heat kernel on ${\bf H}^N$. Then $Γ(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.
The Dirichlet eigenvalue problems for some concave elliptic Hessian operators
本文研究一类广泛凹椭圆Hessian算子的Dirichlet特征值问题，在温和约束下证明了首个非零特征值及特征函数的存在性与正则性，并推广至Gårding-Dirichlet算子，建立了分岔理论。
Strict Log-Concavity of First Dirichlet Eigenfunctions under Symmetry in $\mathbb{CP}^n$
本文证明在复射影空间中，具有凸边界且对称的光滑域的第一 Dirichlet 本征函数在低维情形下严格对数凹。
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…
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…
Surface concentration of transmission eigenfunctions
本文研究透射本征函数的几何集中现象，利用广义Weyl定律与耦合边界层–位势算子的遍历性质，证明其在两种意义下集中于区域边界，为相关数值发现提供理论依据。
We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $λ$ is a frequency and the size of the neigborhood is $\mathcal{O}(λ^{-δ})$, then new sharp estimates are established when $δ\ge 1$, while for $0\le δ\le 1/2$, Sogge's estimates turn out to be optimal. In the intermediate region $1/2<δ<1$, we sometimes get sharp estimates as well. Our arguments follow closely a recent work by Burq and Zuily.
We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.
The critical case for the concentration of eigenfunctions on singular Riemannian manifolds
本文研究带边界的奇异黎曼流形上Laplace-Beltrami算子特征函数的集中现象，在临界情形下证明其平均密度分布于所有长度尺度，并给出精确描述。
The Brunn-Minkowski inequality for the first eigenvalue of the Ornstein-Uhlenbeck operator and log-concavity of the relevant eigenfunction
本文证明 Ornstein-Uhlenbeck 算子第一 Dirichlet 特征值关于 Minkowski 加法具有凸性，刻画了部分凸集类中的等号情形，并证明凸区域上相应正特征函数是对数凹的。
On the concentration of the Fourier coefficients for products of Laplace-Beltrami eigenfunctions on real-analytic manifolds
本文研究了闭解析流形上 Laplace-Beltrami 特征函数乘积的 Fourier 系数，证明了在特征值满足特定条件时系数呈指数衰减，并指出仅需有限个系数即可恢复 99% 的 $L^2$-质量。
This work is concerned with operators of the type A = --c$Δ$ acting in domains $Ω$ := $Ω$ ' x (0, H) $\subseteq$ R^d x R ^+. The diffusion coefficient c > 0 depends on one coordinate y $\in$ (0, H) and is bounded but may be discontinuous. This corresponds to the physical model of ''layered media'', appearing in acoustics, elasticity, optical fibers... Dirichlet boundary conditions are assumed. In general, for each $ε$ > 0, the set of eigenfunctions is divided into a disjoint union of three subsets : Fng (non-guided), Fg (guided) and Fres (residual). The residual set shrinks as $ε$ $\rightarrow…
We work on the boundary $\partial M_τ$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_τ)$. We compute $λ\to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_τ = \exp tΞ_{\sqrtρ}$ for the spectral projection kernel $Π_{χ, λ}$ associated to $Π_τD_{\sqrtρ} Π_τ$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_τ$ of…
We consider mass concentration properties of Laplace eigenfunctions $φ_λ$, that is, smooth functions satisfying the equation $-Δφ_λ= λφ_λ$, on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" par…
In this article we develop new techniques for studying concentration of Laplace eigenfunctions $ϕ_λ$ as their frequency, $λ$, grows. The method consists of controlling $ϕ_λ(x)$ by decomposing $ϕ_λ$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $λ^{-\frac{1}{2}}$. We control $ϕ_λ(x)$ by the $L^2$-mass of $ϕ_λ$ on each geodesic tube and derive a purely dynamical statement through which $ϕ_λ(x)$ can be studied. In particular, we obtain estimates on $ϕ_λ(x)$…
Semiclassical localization of Schrödinger's eigenfunctions
本文通过改进的 Landis 猜想方法，建立了 Schrödinger 特征函数在闭合黎曼面上的指数界限，扩展了经典结果至有界势场的情况。
A spectral condition for the control of eigenfunctions of Baouendi-Grushin type operators
本文针对圆柱或环面上的 Baouendi-Grushin 算子，借助变量分离与 Albert-Uhlenbeck 理论，证明谱重数条件可保证特征函数集中不等式，且该条件对次椭圆算子一般成立。
Consider the Dirichlet-Laplacian in $Ω:= (0,L)\times (0,H)$ and choose another open set $ω\subset Ω$. The estimate $0<C_ω\leq R_ω(u):=\frac{\Vert u\Vert^{2}_{L^{2}(ω)}}{\Vert u\Vert^{2}_{L^{2}(Ω)}}\leq \frac{vol(ω)}{vol(ω)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $\forall ω$, the eigenfunctions satisfy $R_{omega}>C_ω>0,\exists ω, ω\not=\emptyset$, such that $\inf R_ω(u)=0$,and we wish to characterize these two sets. For two patterns we give a s…
This is a survey for the JDG 50th Anniversary conference of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are `local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are `global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, givi…
Log-Concavity of First Dirichlet Eigenfunctions on $\mathbb{CP}^2$
本文证明$\mathbb{CP}^2$中光滑1-凸区域的第一Dirichlet特征函数满足定量对数凹估计，从而严格对数凹，并推出水平集凸性与基本间隙下界$46/5$。
For a large class of semiclassical operators $P(h)-z$ which includes Schrödinger operators on manifolds with boundary, we construct the Quantum Monodromy operator $M(z)$ associated to a periodic orbit $γ$ of the classical flow. Using estimates relating $M(z)$ and $P(h)-z$, we prove semiclassical estimates for small complex perturbations of $P(h) -z$ in the case $γ$ is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator constru…
We develop a theory of multidimensional randomization in Lebesgue spaces $L^p$ with the aid of Kahane-Khintchine-Marcus-Pisier inequalities. More precisely, we obtain a result in the spirit of Maurey-Pisier's theorem which involves random matrices and proves that the multidimensional randomization is universal in $L^p$. Then, we deal with the question of studying necessary and sufficient conditions to get the almost sure convergence in $L^p$ of random linear combinations of eigenfunctions in a compact Riemannian manifold (the famous Paley-Zygmund theorem gives the answer for tori). We introduc…
Non-concentration estimates for Laplace eigenfunctions on compact $C^{\infty}$ manifolds with boundary
本文在具有光滑边界的紧致$C^{\infty}$流形上，通过局部方法扩展了拉普拉斯特征函数的非集中估计，并给出了特征函数的上界估计。
Concavity and hot spots in elliptic problems under mixed boundary conditions
研究平面凸曲线扇形区域上混合边界条件的扭转函数和第一拉普拉斯特征函数的凹性，证明在顶点内角不超过π/2时严格(1/2)-凹和严格对数凹，并推导出唯一热点位于Neumann-Neumann顶点。
We consider the problem of proving $L^p$ bounds for eigenfunctions of the Laplacian in the high frequency limit in the presence of nonpositive curvature and more generally, manifolds without conjugate points. In particular, we prove estimates at the "critical exponent" $p_c = \frac{2(d+1)}{d-1}$, where a spectrum of scenarios for phase space concentration must be ruled out. Our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical $L^p$ bounds of the second author.
We prove $L^p$-mass concentration properties of Laplace eigenfunctions away from their nodal sets, extending a recent result in \cite{GM3} to all dimensions, and giving a slight refinement of a result in \cite{JN}. As a consequence, we are able to derive a sharp Wasserstein uncertainty principle that holds uniformly in the high frequency regime, proving a conjecture in \cite{St1}.
Growth estimates on optimal transport maps via concentration inequalities
本文通过集中不等式和映射单调性，为Brenier映射的多项式上界提供替代证明并扩展结果，涵盖从高斯到对数凹测度的二次增长情形。
Power-logconcavity of the Laplacian ground state
研究证明在特定归一化条件下，拉普拉斯基底态的对数函数的平方根具有凹性，拓展了对数凹性理论。
Concentration in selection-mutation models: error estimates and asymptotic expansions
针对描述表型结构化种群演化的选择-突变积分-微分方程，在小突变效应假设下，严格推导解的一阶渐近展开式，建立Hamilton-Jacobi方法所得近似与生物学量的关联。
Mass concentration in rescaled first order integral functionals
本文研究带质量约束的一阶局部极小化问题，证明最小能量函数总是凹的，并证明能量重标度 Γ-收敛到 H-质量，从而揭示 Cahn-Hilliard 流体向液滴集中等现象。
Sharp local $L^p$ estimates for the Hermite eigenfunctions
本文为 Hermite 特征函数建立了尖锐的局部 $L^p$ 估计，推广了 Thangavelu 的结果并改进了 Koch-Tataru 的估计，解释了全局界随 $p$ 减小的现象，并通过构造例子证明了最优性。
Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus
本文研究d维环面上拉普拉斯特征函数关于任意Borel测度的迹不等式与可观测性不等式，基于测度可积性、正则性及支撑集维数给出充分与必要条件，将Zygmund和Bourgain–Rudnick经典结果推广到高维，并应用于量子极限约束与Schrödinger方程控制理论。
Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues
本文研究了Neumann特征函数边界值的上界和下界，并证明了在频谱窗口内该边界数据的近正交性。通过引入一个包含频谱权重的适当范数，这些界是紧致的，不依赖于特征值。此外，作者还改进了Helmholtz参数到最近Neumann特征值的距离的估计，并展示了在计算Neumann特征对时，这种方法在数值实现上的改进。
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