The gap between the first two eigenvalues for Schrödinger operators with Dirichlet–Neumann boundary condition

Abstract

In this paper, the Schrödinger operatorwhere the potential q.x/ is single-well with transition point a D 2 is considered. Suppose f n g n 1 is the set of eigenvalues for the previous Schrödinger operator and the potential q.x/ possesses an additional condition, we show that the first two eigenvalues 2 and 1 satisfy 2 1 2.Equality holds if and only if q.x/ is a constant.