Effective Perturbation Methods for One-Dimensional Schrödinger Operators

If V is in L, virtually any perturbation technique allows one to control u (and, in fact, to show that all solutions of (1) are bounded as x → ∞). We are interested in cases where V is not L but is small at infinity in some sense. We want to generalize what has turned out to be a powerful set of tools in case V0 ≡ 0, namely, the use of modified Prüfer equations and their discrete analogs, which were dubbed EFGP equations in [6] on account of contributions of [4, 5, 11]. Explicitly, in the continuum case when V0 ≡ 0 and E = k > 0, one defines R(x), θ(x) by