Discrete spectrum of Schrödinger operators with oscillating decaying potentials

Abstract

We consider the Schrödinger operator $H_{ηW} = -Δ+ ηW$, self-adjoint in $L^2({\mathbb R}^d)$, $d \geq 1$. Here $η$ is a non constant almost periodic function, while $W$ decays slowly and regularly at infinity. We study the asymptotic behaviour of the discrete spectrum of $H_{ηW}$ near the origin, and due to the irregular decay of $ηW$, we encounter some non semiclassical phenomena. In particular, $H_{ηW}$ has less eigenvalues than suggested by the semiclassical intuition.