WKB Asymptotic Behavior of Almost All Generalized Eigenfunctions for One-Dimensional Schrِdinger Operators with Slowly Decaying Potentials

We prove the WKB asymptotic behavior of solutions of the differential equation −d2u/dx2+V(x) u=Eu for a.e. E>A where V=V1+V2, V1∈Lp(R), and V2 is bounded from above with A=lim supx→∞ V(x), while V′2(x)∈Lp(R), 1⩽p<2. These results imply that Schrödinger operators with such potentials have absolutely continuous spectrum on (A, ∞). We also establish WKB asymptotic behavior of solutions for some energy-dependent potentials.