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 {uλ}λ∈R which satisfy uλ(x) = u0(x) for all x < 0, we prove that the zeros of b(λ) := W [u0, uλ], the Wronskian of u0 and uλ, 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 + λV u = Eu gives rise to a reflection coefficient which vanishes on a set of couplings with an accumulation point, then V ≡ 0.