Let H ( x , p ; V ) = K ( p ) + φ φ ( x ; V ) be the Hamiltonian of a one-dimensional system, where K = p 2 2 m is the kinetic energy and φ φ ( x ; V ) is a "U-shaped" potential energy profile which depends on a parameter V . Let ⟨ ⋅ ⋅ ⟩ t denote the time average. Let