WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrodinger operators with slowly decaying potentials

We prove the WKB asymptotic behavior of solutions of the differential equat ion -d(2)u/dx(2) + V(x) u = Eu for a.e. E > A where V = V-1 + V-2, V-1 is a n element of L-p(R), and V-2 is bounded from above with A = lim sup(x --> i nfinity) V(x), while V-2'(s) is an element of L-p(R), 1 less than or equal to p < 2. These results imply that Schrodinger operators with such potentia ls have absolutely continuous spectrum on (A, <infinity>). We also establis h WKB asymptotic behavior of solutions for some energy-dependent potentials . (C) 2001 Academic Press.