紧流形上的Schrödinger算子的谱间隙估计

数学物理学报 ›› 2020, Vol. 40 ›› Issue (2): 257-270.

• 论文 • 下一篇

紧流形上的Schrödinger算子的谱间隙估计

何跃^1,*(),赫海龙²()

¹ 南京师范大学数学科学学院数学研究所南京 210023
² 暨南大学数学系广州 510632

收稿日期:2018-08-08 出版日期:2020-04-26 发布日期:2020-05-21
通讯作者: 何跃 E-mail:heyue@njnu.edu.cn;hailongher@126.com; hailongher@jnu.edu.cn
作者简介:赫海龙, E-mail:hailongher@126.com, hailongher@jnu.edu.cn
基金资助:
国家自然科学基金(11671209);国家自然科学基金(11871278);江苏高校优势学科建设工程资助项目

An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds

Yue He^1,*(),Hailong Her²()

¹ Institute of Mathematics, School of Mathematics Sciences, Nanjing Normal University, Nanjing 210023
² Department of Mathematics, Jinan University, Guangzhou 510632

Received:2018-08-08 Online:2020-04-26 Published:2020-05-21
Contact: Yue He E-mail:heyue@njnu.edu.cn;hailongher@126.com; hailongher@jnu.edu.cn
Supported by:
the NSFC(11671209);the NSFC(11871278);the Priority Academic Program Development of Jiangsu Higher Education Institutions

摘要/Abstract

摘要：

$M$是一个$n$维紧黎曼流形,具有严格凸边界,且Ricci曲率不小于$(n-1)K$(其中$K\geq0$为某个常数).假定Schrödinger算子的Dirichlet (或Robin)特征值问题的第一特征函数$f_1$在$M$上是对数凹的,该文得到了此类Schrödinger算子的前两个Dirichlet(或Robin)特征值之差的下界估计,这推广了最近Andrews等人在$\mathbb{R} ^n$中有界凸区域上关于Laplace算子的一个相应结果^[4].

关键词: Schrödinger算子, Dirichlet特征值, Robin特征值, 谱间隙, 具有凸边界的流形, Ricci曲率

Abstract:

Let $M$ be an $n$-dimensional compact Riemannian manifold with strictly convex boundary. Suppose that the Ricci curvature of $M$ is bounded below by $(n-1)K$ for some constant $K\geq0$ and the first eigenfunction $f_1$ of Dirichlet (or Robin) eigenvalue problem of a Schrödinger operator on $M$ is log-concave. Then we obtain a lower bound estimate of the gap between the first two Dirichlet (or Robin) eigenvalues of such Schrödinger operator. This generalizes a recent result by Andrews et al. ([4]) for Laplace operator on a bounded convex domain in $\mathbb{R} ^n$.

Key words: Schrödinger operator, Dirichlet eigenvalue, Robin eigenvalue, Spectral gap, Manifold with convex boundary, Ricci curvature

中图分类号:

O186.1

何跃,赫海龙. 紧流形上的Schrödinger算子的谱间隙估计[J]. 数学物理学报, 2020, 40(2): 257-270.

Yue He,Hailong Her. An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds[J]. Acta mathematica scientia,Series A, 2020, 40(2): 257-270.

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紧流形上的Schrödinger算子的谱间隙估计

An Estimate of Spectral Gap for Schrödinger Operators on Compact Manifolds

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