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

✍ Scribed by Michael Christ; Alexander Kiselev

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 192 KB
• Volume: 179
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3688

✦ Synopsis

We prove the WKB asymptotic behavior of solutions of the differential equation &d 2 uÂdx 2 +V(x) u=Eu for a.e. E>A where V=V 1 +V 2 , V 1 # L p (R), and V 2 is bounded from above with A=lim sup x Ä V(x), while V$ 2 (x) # L p (R), 1 p<2. These results imply that Schro dinger operators with such potentials have absolutely continuous spectrum on (A, ). We also establish WKB asymptotic behavior of solutions for some energy-dependent potentials.

2001 Academic Press

to have WKB-type asymptotics for almost every E, with respect to Lebesgue measure, and for the Schro dinger operator &D 2 +V(x) to have absolutely continuous spectrum on the positive semi-axis. Our main interest here is in potentials V which are a combination of potentials that