Subellipticity of some complex vector fields related to Witten Laplacian

Witten-Laplacian的Steklov特征值的Brock型等周不等式

Brock-type isoperimetric inequality for Steklov eigenvalues of the Witten-Laplacian

本文在加权体积固定约束下证明了欧氏与双曲空间中有界域上Witten-Laplacian的Steklov型特征值的Brock型等周不等式，拓展了F. Brock的经典结果，并提出相关开放问题。

Pollicott-Ruelle spectrum and Witten Laplacians

We study the asymptotic behaviour of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectrum of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. In particular, our results relate the approach of Laudenbach and Harvey--Lawson to Morse theory using currents, which was discussed in previous work of the authors, and Witten's point of view based on semiclassical analysis and tunneling. As an application of our methods, we also construct a natural family o…

Small eigenvalues of the Witten Laplacian with Dirichlet boundary conditions: the case with critical points on the boundary

In this work, we give sharp asymptotic equivalents in the limit $h\to 0$ of the small eigenvalues of the Witten Laplacian, that is the operator associated with the quadratic form $$ ψ\in H^1_0(Ω)\mapsto h^2 \int_Ω\big \vert \nabla \big (e^{\frac 1hf} ψ\big )\big \vert^2\, e^{-\frac 2hf},$$where $\overlineΩ=Ω\cup \partial Ω$ is an oriented $C^\infty$ compact and connected Riemannian manifold with non empty boundary $\partial Ω$ and $f: \overline Ω\to \mathbb R$ is a $C^\infty$ Morse function. The function $f$ is allowed to admit critical points on $ \partial Ω$, which is the main novelty of thi…

Witten-Laplacian 的 Dirichlet 与 Neumann 特征值的若干等周不等式

Several isoperimetric inequalities of Dirichlet and Neumann eigenvalues of the Witten-Laplacian

本文利用重排技术与试验函数构造，在固定加权体积条件下建立空间形式中有界区域上 Witten-Laplacian 多个 Dirichlet 与 Neumann 特征值的等周不等式，推广了 Laplacian 的经典结果。

带位势的Witten-Laplace算子相关Dirichlet-to-Neumann映射的谱不变量

Spectral invariants of the Dirichlet-to-Neumann map associated to the Witten-Laplacian with potential

本研究针对带光滑边界的紧连通黎曼流形，提出计算带位势的Witten-Laplace算子相关Dirichlet-to-Neumann映射谱渐近公式全系数的有效方法，显式给出前四个谱不变量，可提供流形体积、曲率等信息。

关于Witten-Laplacian低阶Neumann特征值的Ashbaugh-Benguria型猜想

On the Ashbaugh-Benguria type conjecture about lower-order Neumann eigenvalues of the Witten-Laplacian

本文证明欧氏与双曲空间中有界域上Witten-Laplacian低阶非零Neumann特征值的等周不等式，强化了Szegő-Weinberger不等式，涵盖Xia-Wang关于Ashbaugh-Benguria猜想的进展，并提出两个推广该猜想的开放问题。

Witten deformation for noncompact manifolds with bounded geometry

Motivated by the Landau-Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function $f$ near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of Thom-Smale complex of $f$ as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large $T$. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomol…

A Thom-Smale-Witten theorem on manifolds with boundary

Given a smooth compact manifold with boundary, we show that the subcomplex of the deformed de Rham complex consisting of eigenspaces of small eigenvalues of the Witten Laplacian is canonically isomorphic to the Thom-Smale complex constructed by Laudenbach. Our proof is based on Bismut-Lebeau's analytic localization techniques. As a by-product, we obtain Morse inequalities for manifolds with boundary.

Analytic eigenbranches in the semi-classical limit

We consider a one parameter family of Laplacians on a closed manifold and study the semi-classical limit of its analytically parametrized eigenvalues. Our results are analogous to a theorem for scalar Schrödinger operators on Euclidean space by Luc Hillairet and apply to geometric operators like Witten's Laplacian associated with a Morse function.

Metastable diffusions with degenerate drifts

We study the spectrum of the semiclassical Witten Laplacian $Δ_{f}$ associated to a smooth function $f$ on ${\mathbb R}^d$. We assume that $f$ is a confining Morse--Bott function. Under this assumption we show that $Δ_{f}$ admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring-Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.

半经典热核渐近与Morse不等式

Semi-classical Heat Kernel Asymptotics and Morse Inequalities

本文通过引入半经典分析中的局部化和标度技术，研究了Witten Laplacian热核在Morse函数临界点附近的渐近行为，并利用这些渐近结果给出了Morse不等式的新颖证明。