On the semiclassical approximation to the eigenvalue gap of Schrِdinger operators

We consider two types of Schrödinger operators H ( t ) = − d 2 / d x 2 + q ( x ) + t cos x and H ( t ) = − d 2 / d x 2 + q ( x ) + A cos ( t x ) defined on L 2 ( 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 λ 1 ( t ) and λ 2 ( t ) the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap Γ ( t ) = λ 2 ( t ) − λ 1 ( t ) in the limit as t → ∞ .