WKB Asymptotic Behavior of Almost All Generalized Eigenfunctions for One-Dimensional Schrödinger Operators with Slowly Decaying Potentials
By Michael Christ, Alexander V. Kiselev. 2001
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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.
Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/2bba5b9f62563a9ed58cf57b889a1436fea5b369
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In Atomic and Molecular Physics, and Optics
In Mathematical Physics
Citations in the corpus
- Multiple Vector Valued Inequalities via the Helicoidal Method (2015, methodology): “21) M(f 1;f 2;g)(˘) = x 1<x 2<x 3 f^ 1(x 1)f^ 2(x 2)g(x 3)e2ˇi˘(x 1+x 2+x 3)dx 1dx 2dx 3; where f 1 2Lp 1;f 2 2Lp 2, 1 <p 1;p 2 <1, and g2Lpwith 1 <p<2. The techniques from [CK01a], [CK01b],[CK98], akin to those used by Paley in [Pal31], are based on a dyadic ltration associated to”
- A Carleson type theorem for a Cantor group model of the scattering transform (2002, background): “Beals and Coifman [2] study in detail the case when the potential is (generic) in L1 or in weighted spaces L1 ∩ L2 with weights of the form (1 + |x|)m for suitable m.”
- Lp estimates for the biest II. The Fourier case (2001, methodology): “We became interested in this operator while studying WKB expansions of eigenfunctions of one-dimensional Schr¨ odinger operators, following the work of Christ and Kiselev [ 1 ], [2].”