Laplacian算子的旋转奇异摄动

薄平面区域上拉普拉斯算子的Dirichlet特征值和特征函数的奇异渐近展开

Singular asymptotic expansions for Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin planar domains

我们考虑拉普拉斯算子的Dirichlet边界条件的平面域和研究的效果，在一个方向上进行缩放的频谱。我们推导出本征值和相应的本征函数作为标度参数在零附近的函数的渐近展开。这种方法允许我们，例如，以获得一个近似的第一Dirichlet特征值的一大类平面域，在非常温和的假设。

双曲曲面上Laplacian奇异摄动的迹公式

The trace formula for singular perturbations of the Laplacian on hyperbolic surfaces

本文证明了有限体积双曲曲面上δ势的Selberg迹公式的一个类似形式。为了简单起见，我们限制自己的表面，最多有一个尖点，但我们的方法可以很容易地扩展到任何数量的尖点。在非紧曲面的情况下，我们推导出了Maass尖点形式、剩余Maass形式和非全纯Eisenstein级数的微扰类似物。后者满足一个功能方程在经典的情况下。我们还介绍了Selberg的zeta函数的微扰模拟，并应用迹公式证明其亚纯延拓复平面以及一个功能方程。

Residue families, singular Yamabe problems and extrinsic conformal Laplacians

Let $(X,g)$ be a compact manifold with boundary $M^n$ and $σ$ a defining function of $M$. To these data, we associate natural conformally covariant polynomial one-parameter families of differential operators $C^\infty(X) \to C^\infty(M)$. They arise through a residue construction which generalizes an earlier construction in the framework of Poincaré-Einstein metrics. The main ingredient of the definition of residue families are eigenfunctions of the Laplacian of the singular metric $σ^{-2}g$. We prove that if $σ$ is an approximate solution of a singular Yamabe problem, these families can be wr…

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。

关于具有频繁交替型边界条件的Laplacian模型边值问题

On a model boundary value problem for Laplacian with frequently alternating type of boundary condition

研究了一类具有频繁交替边界条件的二维Laplacian奇异摄动特征值问题。构造了特征元的完全双参数渐近性。

Fractional powers and singular perturbations of quantum differential Hamiltonians

We consider the fractional powers of singular (point-like) perturbations of the Laplacian, and the singular perturbations of fractional powers of the Laplacian, and we compare such two constructions focusing on their perturbative structure for resolvents and on the local singularity structure of their domains. In application to the linear and non-linear Schrödinger equations for the corresponding operators we outline a programme of relevant questions that deserve being investigated.

Quantum confinement for the curvature Laplacian $-Δ+cK$ on 2D-almost-Riemannian manifolds

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set $Z$, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Be…

Small-time bilinear control of Schrödinger equations with application to rotating linear molecules

In [14] Duca and Nersesyan proved a small-time controllability property of nonlinear Schrödinger equations on a d-dimensional torus $\mathbb{T}^d$. In this paper we study a similar property, in the linear setting, starting from a closed Riemannian manifold. We then focus on the 2-dimensional sphere $S^2$, which models the bilinear control of a rotating linear top: as a corollary, we obtain the approximate controllability in arbitrarily small times among particular eigenfunctions of the Laplacian of $S^2$.

2-球面上带三个锥形奇点的常曲率度量的拉普拉斯算子行列式

Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

该研究推导了带三个锥形奇点的2-球面常曲率度量下Friedrichs拉普拉斯算子zeta正则化谱行列式的显式公式，证明了对称奇点对应行列式驻点（小表面积时为极小值），并建立了其与刘维尔场论中刘维尔作用量的关联。