Absence of reflection as a function of the coupling constant

We consider solutions of the one-dimensional equation - u″+(Q+λV)u=0 where Q:ℝ→ℝ is locally integrable, V:ℝ→ℝ is integrable with supp(V) ⊂[0, 1], and λ ∈ ℝ is a coupling constant. Given a family of solutions {u _λ}_λ∈ℝ which satisfy u _λ(x) = u₀(x) for all x<0, we prove that the zeros of b(λ) := W[u₀, u_λ], the Wronskian of u₀ and u_λ, form a discrete set unless V≡0. Setting Q(x) :=-E, one sees that a particular consequence of this result may be stated as: if the fixed energy scattering experiment -u″+λVu=Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V≡0.