Schrödinger operators with highly singular oscillating potentials

Abstract

We investigate the Feynman-Kac semigroupP ^V_t and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator −1/2Δ+V on ℝ^d\{0}.V will be a highly singular, oscillating potential like

$$V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$$

with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ℝ such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at.

On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.