Here V ± ± is always positive and the problem is one of reflection and transmission of waves incident from r ∗ ∗ → → ∞ ∞ to r ∗ ∗ → → − − ∞ ∞ . The problem is essentially the same as that of a reflection and transmission problem by a potential barrier in quantum mechanics. Let the incident wave with unit amplitude be e + i σ σ r ∗ ∗ , then the asymptotic behaviours of the solution are given by