Schrödinger Operators with Rapidly Oscillating Potentials

Abstract.

Schrödinger 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 Schrödinger 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.