Weyl law for semi-classical resonances with randomly perturbed potentials

随机扰动弹性杆力学性质的高效不确定性量化

Efficient uncertainty quantification for mechanical properties of randomly perturbed elastic rods

针对随机几何扰动下弹性杆有效模量的不确定性量化问题，本文通过细长杆Γ-极限推导一维替代模型，建立随机均匀化结果，并结合有限元与蒙特卡罗模拟验证其有效性。

Acoustic and geoacoustic inverse problems in randomly perturbed shallow-water environments

The main goal of this paper is to estimate the regional acoustic and geoacoustic shallow-water environment from data collected by a vertical hydrophone array and transmitted by distant time-harmonic point sources. We aim at estimating the statistical properties of the random fluctuations of the index of refraction in the water column and the characteristics of the sea bottom. We first explain from first principles how acoustic wave propagation can be expressed as Markovian dynamics for the complex mode amplitudes of the sound pressure, which makes it possible to express the cross moments of th…

Refined Weyl law for homogeneous perturbations of the harmonic oscillator

Let $H$ denote the harmonic oscillator Hamiltonian on $\mathbb{R}^d,$ perturbed by an isotropic pseudodifferential operator of order $1.$ We consider the Schrödinger propagator $U(t)=e^{-itH},$ and find that while $\operatorname{singsupp} \operatorname{Tr} U(t) \subset 2 π\mathbb{Z}$ as in the unperturbed case, there exists a large class of perturbations in dimension $d \geq 2$ for which the singularities of $\operatorname{Tr} U(t)$ at nonzero multiples of $2 π$ are weaker than the singularity at $t=0$. The remainder term in the Weyl law is of order $o(λ^{d-1})$, improving in these cases the $…

具有奇异势的尖锐 Weyl 定律

Sharp Weyl laws with singular potentials

本文研究三维 Riemann 流形上受 Kato 类奇异势扰动的 Laplace–Beltrami 算子，推广 Avakumović 方法，证明带附加项和尖锐余项的逐点 Weyl 定律，并构造反例说明附加项可破坏标准尖锐定律。

The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction $A_B$ of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces $\Sc$ and $\tilde{\Sc}$ such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples -- one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary diff…

Ergodicity for the randomly forced Navier-Stokes system in a two-dimensional unbounded domain

The ergodic properties of the randomly forced Navier-Stokes system have been extensively studied in the literature during the last two decades. The problem has always been considered in bounded domains, in order to have, for example, suitable spectral properties for the Stokes operator, to ensure some compactness properties for the resolving operator of the system and the associated functional spaces, etc. In the present paper, we consider the Navier-Stokes system in an unbounded domain satisfying the Poincaré inequality. Assuming that the system is perturbed by a bounded non-degenerate noise,…

Toeplitz band matrices with small random perturbations

We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of eigenvalues in certain domains, which may depend on $N$, with probability sub-exponentially (in $N$) close to $1$. We show that most eigenvalues of the perturbed Toeplitz matrix are at a distance of at most $\mathcal{O}(N^{-1+\varepsilon})$, for all $\varepsilon >0$, to the curve in the complex plane given by the symbol of the unperturbed Toeplitz matrix.

Ergodicity in Randomly Forced Rayleigh-Bénard Convection

We consider the Boussinesq approximation for Rayleigh-Bénard convection perturbed by an additive noise and with boundary conditions corresponding to heating from below. In two space dimensions, with sufficient stochastic forcing in the temperature component and large Prandtl number, we establish the existence of a unique ergodic invariant measure. In three space dimensions, we prove the existence of a statistically invariant state, and establish unique ergodicity for the infinite Prandtl Boussinesq system. Throughout this work we provide streamlined proofs of unique ergodicity which invoke an …

Hardy's uncertainty principle and unique continuation property for stochastic heat equations

The goal of this paper is to prove a uniqueness result for a stochastic heat equation with a randomly perturbed potential, which can be considered as a variant of Hardy's uncertainty principle for stochastic heat evolutions.

奇异扰动算子预解式差的谱性质

Spectral properties of the resolvent difference for singularly perturbed operators

本文研究了奇异扰动椭圆算子预解式差的谱估计问题，在测度满足Ahlfors条件及权函数满足积分条件下获得了阶锐估计，并证明了特征值的Weyl型渐近性。