Discrete spectrum for Schrodinger operators with oscillating decaying potentials

We consider the Schrodinger operator H(eta)w = -Delta + eta W, self-adjoint in L-2 (R-d), d >= 1. Here eta is a non-constant oscillating function, while W decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of H(eta)w near the origin, and due to the irregular decay of eta W, we encounter some non-semiclassical phenomena. In particular, H(eta)w has less eigenvalues than suggested by the semiclassical intuition. (C) 2016 Elsevier Inc. All rights reserved.