紧流形上的Schrodinger算子的谱间隙估计

An Estimate of Spectral Gap for Schrodinger Operators on Compact Manifolds

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≥0为某个常数).假定Schrodinger算子的Dirichlet (或Robin)特征值问题的第一特征函数f1在M上是对数凹的,该文得到了此类Schrodinger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R^n中有界凸区域上关于Laplace算子的一个相应结果[4].

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≥0 and the first eigenfunction f1 of Dirichlet(or Robin)eigenvalue problem of a Schrodinger 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 Schrodinger operator.This generalizes a recent result by Andrews et al.([4])for Laplace operator on a bounded convex domain in R^n.