WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrödinger operators with slowly decaying potentials (Q5927519)

The main result of this paper gives asymptotics for solutions \(u_\pm(x,E)\) to \(-u''+(V-E)u=0\), where \(V=V_1+V_2\) on \(\mathbb{R}\), with (i) \(V_2\) bounded; (ii) \(V_1\) and \(V'_2\in \ell^p(L^1)(\mathbb{R})\) for some \(p\in[1,2)\) (which means that their \(L^1\) norms on \([n,n+1]\) form an \(\ell^p\) sequence). The authors find that, for almost all \(E>\limsup_{x\to\infty}V_2(x)\), there are solutions with ``WKB asymptotics''. A part of the proof relies on a paper by the authors, which is to appear in the same journal.