The gap between the first two eigenvalues of Schrödinger operators with single-well potential

In this paper the Schrödinger operators endowed with Neumann or Dirichlet boundary conditions, where the potential q ∈ L1[0, π] is single-well with transition point a ∈ (0, π), are considered. Suppose {νn}n ≥ 0 and {λn}n ≥ 1 are the set of eigenvalues for Neumann or Dirichlet problem, respectively, and the potential q(x  ) possesses an additional condition, we present estimates of lower bound for the differences ν1−ν0ν1−ν0 and λ2−λ1,λ2−λ1, respectively. The results show that the energy transition for quantum particles jumping to the ground state is related to transition point a.