一类稀疏树上Laplacian的奇异连续谱

Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the …

具有Dehn手术型奇点的双曲三维流形上的Laplacian

The Laplacian on hyperbolic 3-manifolds with Dehn surgery type singularities

我们研究了具有Dehn手术型奇异性的双曲三维流形上的拉普拉斯算子的谱及其对广义Dehn手术系数的依赖性。

通过Belyi函数和拉普拉斯算子行列式研究奇异常曲率球面的三角剖分

Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

本文通过Belyi函数和拉普拉斯行列式研究奇异常曲率球面的三角剖分，并计算了不同正多面体的行列式。

有限维稀疏树的奇异连续稠密点谱

Singular Continuous and Dense Point Spectrum for Sparse Trees with Finite Dimensions

稀疏树是具有稀疏分支的树。这些树中的一些上的拉普拉斯算子可以被证明具有奇异的谱测度。我们专注于一个简单的家庭的稀疏树的维度可以自然地定义和有限的。一般地，这个族具有奇异的谱测度和在某个区间上稠密的特征值。

Sparse estimation of Laplacian eigenvalues in multiagent networks

We propose a method to efficiently estimate the Laplacian eigenvalues of an arbitrary, unknown network of interacting dynamical agents. The inputs to our estimation algorithm are measurements about the evolution of a collection of agents (potentially one) during a finite time horizon; notably, we do not require knowledge of which agents are contributing to our measurements. We propose a scalable algorithm to exactly recover a subset of the Laplacian eigenvalues from these measurements. These eigenvalues correspond directly to those Laplacian modes that are observable from our measurements. We …

Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition

We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behavi…

奇异Dirichlet分数Laplacian算子基态的点态估计

Pointwise estimates for the ground states of singular Dirichlet fractional Laplacian

在相对紧的欧氏子集上建立了奇异分数阶薛定谔算子基态的精确逐点估计。所考虑的算子是$(-Δ)^{α/2}|_\Om-c|x|^{-α}$型的，其中$(-Δ)^{α/2}|_\Om$是$\R$中具有零外部条件的开子集$\Om$上的分数Laplacien，而$0<c\leq(\frac{d-α}{2})^2$是开子集$0<c\leq(\frac{d-α}{2})^2$上的分数Laplacien.讨论了这类算子的内禀超压缩性，并得到了热核的一个大时间渐近解。

奇异斯图尔-刘维勒算子的谱渐进行为及其在磁拉普拉斯算子和库默函数a零点中的应用

Spectral asymptotics for a class of singular Sturm-Liouville operators with applications to magnetic Laplacian and a-zeros of Kummer functions

本文研究了一类奇异斯图尔-刘维勒算子在半经典极限下的谱底行为，通过与库默和惠特AKER函数的联系，推导了这些函数非渐近零点关于第一个参数的精确定位结果，并探讨了磁拉普拉斯算子在强磁场极限下的谱特性。

On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisa…

带位势项的奇异p-Laplacian方程的极小增长正解

On positive solutions of minimal growth for singular p-Laplacian with potential term

令$Ω$为$\mathbb{R}^d$、$d\geq 2$和$1<p<\infty$中的结构域。修复$V\in L_{\mathrm{loc}}^\infty(Ω)$。考虑由Q（u）给出的函数$Q$及其Gâteaux导数$Q^\prime$：= \frac{1}{p} \int_Ω（|\nabla u| ^p+V |u| ^p）\dx，Q^ \prime（u）：=- \nabla \cdot（|\nabla u| ^{p-2} \nabla u）+V |u| ^{p-2}u。假设$Q\geq 0$在$C_0^\infty(Ω)$上。在以前的文章中，我们讨论了$C_0^\infty(Ω)$上的泛函$Q$的弱极性的缺乏和广义基态的存在之间的关系。本文进一步研究了泛函$Q$的泛函分析性质与方程$Q^\prime (u)=0$正解性质之间的关系。

Spectral asymptotics for sub-Riemannian Laplacians

We study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration …

Laplacians on smooth distributions

Let $M$ be a compact smooth manifold equipped with a positive smooth density $μ$ and $H$ be a smooth distribution endowed with a fiberwise inner product $g$. We define the Laplacian $Δ_H$ associated with $(H,μ,g)$ and prove that it gives rise to an unbounded self-adjoint operator in $L^2(M,μ)$. Then, assuming that $H$ generates a singular foliation $\mathcal F$, we prove that, for any function $φ$ from the Schwartz space $\mathcal S(\mathbb R)$, the operator $φ(Δ_H)$ is a smoothing operator in the scale of longitudinal Sobolev spaces associated with $\mathcal F$. The proofs are based on pseudo…

具有kinks的流形及其高斯核图拉普拉斯算子的渐进行为

Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel

本文研究了具有kinks的流形上高斯核图拉普拉斯算子的渐进行为，推导了其在内部和奇点附近的特性，并通过数值模拟验证了理论结果。