Jens Marklof, Zeev Rudnick
We consider an orthonormal basis of eigenfunctions of the Dirichlet Laplacian for a rational polygon. The modulus squared of the eigenfunctions defines a sequence of probability measures. We prove that this sequence contains a density-one subsequence that converges to Lebesgue measure.
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Quantum mixing for eigenfunctions of rational polygons in configuration space
本文扩展了有理多边形本征函数的均匀分布结果至非对角元，还给出2-环面相同结果的另一证明，依托方向台球流的弱混合性开展研究。
Riemannian Manifolds With Uniformly Bounded Eigenfunctions
平坦环面$\R^n / L$上的标准本征函数$ϕ_λ = e^{i < λ, x >}$具有独立于本征值有界的$L^{\infty}$ -范数.在无理平坦环面的情况下，$L^2$ -正规化的本征函数具有一致有界的$L^{\infty}$ -范数.类似的基也存在于其他平坦流形上。这个性质是否是平坦流形的特征？对于具有完全可积测地线流的紧致黎曼流形，我们给出了肯定的回答。
We exhibit limit-periodic Schrödinger operators that are uniformly localized in the strongest sense possible. That is, for these operators there are uniform exponential decay rates such that every element of the hull has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical localization.
We study a random Schroedinger operator, the Laplacian with N independently uniformly distributed random delta potentials on flat tori T^d_L = R^d/LZ^d, d = 2, 3, where L > 0 is large. We determine a condition in terms of the size of the torus L, the density of the potentials ρ= N/L^d and the energy of the eigenfunction E such any such eigenfunctions will with nonzero probability be uniformly distributed on the entire torus. We remark that the equidistribution we prove here is still consistent with a localized regime, where the localization length is much larger than the size of the torus. In …
We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large number of sides), we show that these fluctuations are asymptotically governed by a Gaussian free (massless) field. This complements the similar result obtained in Kenyon [Comm. Math. Phys. 281 (2008) 675-709] about tilings of regions without frozen facets of the limit shape. In our asymptotic analysis we use the explicit double contour integral formula for the determinantal correlatio…
In this note we show that if a family of ergodic Schrödinger operators on $l^2({\Bbb Z}^γ)$ with continuous potentials have uniformly localized eigenfunctions then these eigenfunctions must be uniformly localized in a homogeneous sense.
We study existence and absence of $\ell^2$-eigenfunctions of the combinatorial Laplacian on the 11 Archimedean tilings of the Euclidean plane by regular convex polygons. We show that exactly two of these tilings (namely the $(3.6)^2$ "Kagome" tiling and the $(3.12^2)$ tiling) have $\ell^2$-eigenfunctions. These eigenfunctions are infinitely degenerate and are constituted of explicitly described eigenfunctions which are supported on a finite number of vertices of the underlying graph (namely on the hexagons and $12$-gons in the tilings, respectively). Furthermore, we provide an explicit express…
Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist 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 …
We consider a system of self-avoiding polygons interacting through a potential that penalizes or rewards the number of mutual touchings and we provide an exact computation of the critical curve separating a regime of long polygons from a regime of localized polygons. Moreover, we prove the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and we provide a non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments.
In this paper we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show un…
We study the Laplacian perturbed by two delta potentials on a two-dimensional flat torus. There are two types of eigenfunctions for this operator: old, or unperturbed eigenfunctions which are eigenfunctions of the standard Laplacian, and new, perturbed eigenfunctions which are affected by the scatterers. We prove that along a density one sequence, the new eigenfunctions are uniformly distributed in configuration space, provided that the difference of the scattering points is Diophantine.
We consider Laplacian eigenfunctions in circular, spherical and elliptical domains in order to discuss three kinds of high-frequency localization: whispering gallery modes, bouncing ball modes, and focusing modes. Although the existence of these modes was known for a class of convex domains, the separation of variables for above domains helps to better understand the "mechanism" of localization, i.e. how an eigenfunction is getting distributed in a small region of the domain, and decays rapidly outside this region. Using the properties of Bessel and Mathieu functions, we derive the inequalitie…
We discuss new results on the geometry of eigenfunctions in disor- dered systems. More precisely, we study tori $R^d/LZ^d$, $d=2,3$, with uniformly distributed Dirac masses. Whereas at the bottom of the spectrum eigenfunctions are known to be localized, we show that for sufficiently large eigenvalue there exist uniformly distributed eigenfunctions with positive probability. We also study the limit $L\to\infty$ with a positive density of random Dirac masses, and deduce a lower polynomial bound for the localization length in terms of the eigenvalue for Poisson distributed Dirac masses on $R^d$ .…
Eigenvalues and eigenfunctions of the fractional Laplacian on the interval
该研究针对区间上的分数阶拉普拉斯算子，证明了其本征值的三项渐近公式及归一化本征函数的一致有界性，改进了现有渐近理论并解决了相关猜想。
Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble
本文证明在极限形状技术假设下，单连通多边形随机菱形铺砌在北极边界非奇异点附近的边缘统计收敛于Airy线系综，并利用保持高度函数集中估计的Markov链完成与六边形铺砌的局部比较。
We consider Hermitian and symmetric random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y \in Λ\subset \Z^d$, are independent, uniformly distributed random variables if $\abs{x-y}$ is less than the band width $W$, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t\ll W^{d/3}$. We also show that the localization length of an arbitrarily large majority of the eigenvectors is larger than a factor $W^{d/6}$ times the band width. All results are uniform in the size $\absΛ…
We study the bottom of the spectrum of the Anderson Hamiltonian $\mathcal{H}_L := -\partial_x^2 + ξ$ on $[0,L]$ driven by a white noise $ξ$ and endowed with either Dirichlet or Neumann boundary conditions. We show that, as $L\rightarrow\infty$, the point process of the (appropriately shifted and rescaled) eigenvalues converges to a Poisson point process on $\mathbb{R}$ with intensity $e^x dx$, and that the (appropriately rescaled) eigenfunctions converge to Dirac masses located at independent and uniformly distributed points. Furthermore, we show that the shape of each eigenfunction, recentere…
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。
We use the mirror coupling of Brownian motion to show that under a $β\in (0,1)$-dependent Kato type assumption (which is satisfied under a suitable $L^q$-assumption on the electro-magnetic potential, where $q$ depends on $β$ and the dimension $d$) on the possibly nonsmooth electro-magnetic potential, the corresponding magnetic Schrödinger semigroup in $\mathbb{R}$ has a global $L^{p}$-to-$C^{0,β}$ Hölder smoothing property for all $p\in [1,\infty]$, in particular all eigenfunctions are uniformly $β$-Hölder continuous. This result shows that the eigenfunctions of the Hamilton operator of a mole…
Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards
Rudnick-Sarnak的量子唯一遍历性（QUE）猜想是一致双曲测地线流流形上Laplacian的每一个本征函数φ n在半经典极限（本征值E_n ->无穷大）内都是等分布的，即不存在“强痕”。本文利用分段常数检验函数A的矩阵元<phi_n，A_phi_m>，数值研究了具有Dirichlet边界条件的均匀双曲Sinai型平面欧氏台球（“鼓问题”）在前所未有的高E和统计精度下的等分布率.通过收集30000个对角元素（直到水平n ~ 7*10^5），我们发现它们的方差随本征值的幂0.48 +-0.01衰减，接近Feingold-Peres（FP）的估计值的1/2。这与现有研究的结果形成对比，现有研究仅限于E_n小10^2倍。我们在这个系统中找到了QUE的强有力证据。我们还比较了非对角方差，作为距离对角线的函数，对FP的最高精度（0.7%），迄今为止在任何混沌系统。我们概述了用于计算本征函数的有效标度法。
Quantum Mixing for Schrödinger eigenfunctions in Benjamini-Schramm limit
研究在本杰明-施拉姆极限下，Schrodinger算子的本征函数在足够大的谱窗口内呈现量子混合性质，适用于高概率随机超曲面及Bose气体热力学极限中的Hartree单粒子算子。
Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces
研究在局部对称空间序列的Benjamini-Schramm极限下，不变微分算子的本征函数的平均去局域化特性。
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