论文标题
曲率laplacian $-Δ+ck $的量子限制在2D-摩曼尼亚歧管上
Quantum confinement for the curvature Laplacian $-Δ+cK$ on 2D-almost-Riemannian manifolds
论文作者
论文摘要
步骤2的二维几乎是利马尼亚结构是Grushin平面的自然概括。它们是通用的riemannian结构，局部正交框架的向量可以平行。在2步假设下，单数$ z $，该结构不是riemannian，是1D嵌入的子手机。在接近奇异集的同时，所有riemannian数量都不同。这些结构的一个非凡特性是，地球化学可以在没有奇异性的情况下穿越奇异集，但是Schrödinger方程的热量和解决方案（带有Laplace-Beltrami操作员$δ$）。这是由于以下事实：（在自然的紧凑性假设下），Laplace-Beltrami操作员本质上是在没有奇异集合的歧管的连接组件上进行的自相关。在文献中，这种现象称为量子限制。 在本文中，我们研究了曲率laplacian的自动相关性，即$-Δ+ck $，对于$ c \ in（0,1/2）$（这里$ k $是高斯曲率），该曲率起源于无坐标的量化程序（例如，以途径或协方差或协方差的Weyl量化为例）。我们证明，这种类型的操作员没有量子限制。
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-Beltrami operator $Δ$) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely $-Δ+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.
