On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators H(t)=−d2/dx2+q(x)+tcosx and H(t)=−d2/dx2+q(x)+Acos(tx) defined on L2(R), where q is an even potential that is bounded from below, A is a constant, and t>0 is a parameter. We assume that H(t) has at least two eigenvalues below its essential spectrum; and we denote by λ1(t) and λ2(t) the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap Γ(t)=λ2(t)−λ1(t) in the limit as t→∞.