Schrödinger Operators with Rapidly Oscillating Potentials

Abstract

Schrodinger operators \(\hat{H} = -\Delta + V\) with rapidly oscillating potentials V such as \(cos |x|^{2}\) are considered. Such potentials are not relatively compact with respect to the free Schrodinger operator −Δ. We show that the oscillating potential V do not change the essential spectrum of −Δ. Moreover we derive upper bounds for negative eigenvalue sums of Ĥ.