Hardy inequality and fractional Leibnitz rule for perturbed Hamiltonians on the line

Abstract

We consider the following perturbed Hamiltonian H = −∂ 2x + V (x) on the real line. The potential V (x), satisfies a short range assumption of typeWe study the equivalence of classical homogeneous Sobolev type spaces Ḣs p (R), p ∈ (1, ∞) and the corresponding perturbed homogeneous Sobolev spaces associated with the perturbed Hamiltonian. It is shown that the assumption zero is not a resonance guarantees that the perturbed and unperturbed homogeneous Sobolev norms of order s = γ −1 ∈ [0, 1/p) are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Sobolev spaces of order s ∈ [0, 1/p) invariant.