LOWER BOUNDS FOR TNE FIRST GAP OF EIGENVALUES IN THE SCHRDINGER OPERATOR ON SIMPLE RIEMANNIAN MANIFOLDS

<正> We give a lower bound for the first gap λ2—λ1 of the twolowerst eigenvalues of the Schr（o|¨）dinger operator-△+W（p） with the Dirichletboundary condition and a strictly convex potential W（p）on M in which M is acompact simple Riemannian manifold with smooth strictly convex boundary （?）MHere a compact Riemannian manifold M is said to be simple if M～（?）M istopologically R2.We prove thatλ2-λ1≥（π2）/（d2）+min{0,-（n-1）K}where d is the diameter of M and-（n-1）K,（K≥0）the lower bound of theRicci curvature of M.This work generalizes the results in the classical Eucli-dean situation due to Singer,Wong and Yau,Yu and Zhong to a kind of curvedRiemannian manifold.