Michael Hitrik
For a class of non-selfadjoint semiclassical operators in dimension one, we get a complete asymptotic description of all eigenvalues near a critical value of the leading symbol of the operator on the boundary of the pseudospectrum.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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 consider the semiclassical asymptotic behaviour of the number of eigenvalues smaller than $E$ for elliptic operators in $L\sp 2 ({\bf R}\sp d)$. We describe a method of finding remainder estimates related to the volume of the region of the phase space in which the principal symbol takes values belonging to the interval $[E'-h; E'+h]$, where $E'$ is close to $E$. This method allows to derive remainder estimates $O(h\sp {1-d})$ for a class of symbols with critical points and non-smooth coefficients.
This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal-Bargmann and Hardy spaces associated with the magnetic field.
We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead of the operators. We also utilize some important ideas and technical elements from Helffer and Nier (2006), who were the first to supply a complete proof of the full semi-classical asymptotic expansions for the eigenvalues with fixed numbers.
Boundary spectral estimates for semiclassical Gevrey operators
该研究针对具Gevrey-$s$正则性象征的半经典伪微分算子，推导了主象征值域边界附近的谱与预解式估计，明确了无谱区域尺度与预解式上界，补充了光滑/解析情形对应结果的Gevrey类版本。
Let $Ω\subset \mathbb{R}^d$ be bounded with $C^1$ boundary. In this paper we consider Schrödinger operators $-Δ+ W$ on $Ω$ with $W(x)\approx\mathrm{dist}(x, \partialΩ)^{-2}$ as $\mathrm{dist}(x, \partialΩ)\to 0$. Under weak assumptions on $W$ we derive a two-term asymptotic formula for the sum of the eigenvalues of such operators.
Semiclassical analysis of a nonlocal boundary value problem related to magnitude
本文利用伪微分边界问题技术研究流形上与分数阶拉普拉斯算子相关的 Dirichlet 问题，并在半经典极限下用曲率不变量给出幅度（magnitude）的渐近展开。
We introduce the periodic Airy-Schrödinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a semiclassical regime in which the results are stated for a fixed value of the semiclassical parameter and are thus estimates instead of asymptotic results. We prove that there exists a sequence of explicit constants, which are zeroes of classical functions, giving upper bounds of the semiclassical parameter for which the spectral bands are in the semiclassical regime. We…
We consider the fractional Laplacian on a domain and investigate the asymptotic behavior of its eigenvalues. Extending methods from semi-classical analysis we are able to prove a two-term formula for the sum of eigenvalues with the leading (Weyl) term given by the volume and the subleading term by the surface area. Our result is valid under very weak assumptions on the regularity of the boundary.
We consider the Schrödinger operator $H_{ηW} = -Δ+ ηW$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $η$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{ηW}$ near the origin, and due to the irregular decay of $ηW$, we encounter some non semiclassical phenomena. In particular, $H_{ηW}$ has less eigenvalues than suggested by the semiclassical intuition.
The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in $\mathbb{R}^3$ confined by a local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.
Universality for free fermions and the local Weyl law for semiclassical Schrödinger operators
本文研究了半经典极限下薛定谔算子谱投影算子的局部渐近性，证明了其重标度积分核向普适模型的收敛，并由此推导出自由费米子点过程在微观、宏观和介观尺度的涨落普适性及中心极限定理。
Uniform observation of semiclassical Schr{ö}dinger eigenfunctions on an interval
本文研究区间上单阱势与 Dirichlet 边界条件下的半经典 Schrödinger 特征函数，利用 Agmon 估计、Gronwall 论证及半经典测度，建立在半经典和高能极限下均一致且最优的 L^2 密度上下界。
We prove uniform $L^p$ resolvent estimates for the stationary damped wave operator. The uniform $L^p$ resolvent estimates for the Laplace operator on a compact smooth Riemannian manifold without boundary were first established by Dos Santos Ferreira-Kenig-Salo and advanced further by Bourgain-Shao-Sogge-Yao. Here we provide an alternative proof relying on the techniques of semiclassical Strichartz estimates. This approach allows us also to handle non-self-adjoint perturbations of the Laplacian and embeds very naturally in the semiclassical spectral analysis framework.
Harmonic Approximation and Resolvent Estimates for Semiclassical Non-Self-Adjoint Operators
研究一类具有临界点的非自伴非椭圆h-伪微分算子的预解估计和低端谱界，通过主象征实部沿虚部哈密顿流的平均动力学条件，在半经典伪谱边界附近建立精确预解估计。
We study the spectral properties of Schrödinger operators on a compact connected Riemannian manifold $M$ without boundary in case that the underlying Hamiltonian system possesses certain symmetries. More precisely, if $M$ carries an isometric and effective action of a compact connected Lie group $G$, we prove a generalized equivariant version of the semiclassical Weyl law with an estimate for the remainder, using a semiclassical functional calculus for $h$-dependent functions and relying on recent results on singular equivariant asymptotics. These results will be used to derive an equivariant …
Semiclassical WKB Problem for the non-self-adjoint Dirac operator
本文研究了非自共轭狄拉克算子的半经典 WKB 问题，探讨了其与聚焦立方 NLS 方程的联系。
Spectral asymptotics of the Neumann Laplacian with variable magnetic field on a smooth bounded domain in three dimensions
本文研究了三维光滑有界区域上Neumann磁Laplacian的半经典谱分析，在变磁场的通用假设下建立了最低本征值的半经典展开，并证明其在半经典极限下变为单重。
Semiclassical $L^p$ quasimode restriction estimates in two dimensions
本文在二维情形下建立了光滑曲线上准模的尖锐 $L^p$ 限制估计，并通过分析曲线与半经典双特征流的接触阶，将其应用于紧黎曼流形上的 Laplace-Beltrami 特征函数及 $\mathbb R^2$ 上的 Hermite 函数。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。