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

  title={WKB Asymptotic Behavior of Almost All Generalized Eigenfunctions for One-Dimensional Schr{\"o}dinger Operators with Slowly Decaying Potentials},
  author={Michael Christ and Alexander V. Kiselev},
  journal={Journal of Functional Analysis},
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 Schrodinger operators with such potentials have absolutely continuous spectrum on (A, ∞). We also establish WKB asymptotic behavior of solutions for some energy-dependent potentials. 

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