Discrete spectrum for Schrödinger operators with oscillating decaying potentials

We consider the Schrödinger operator HηW=−Δ+ηWHηW=−Δ+ηW, self-adjoint in L2(Rd)L2(Rd), d≥1d≥1. Here η is a non-constant oscillating function, while W   decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of HηWHηW near the origin, and due to the irregular decay of ηW  , we encounter some non-semiclassical phenomena. In particular, HηWHηW has less eigenvalues than suggested by the semiclassical intuition.