On homogeneous Besov spaces for 1D Hamiltonians without zero resonance

Abstract

We consider 1-D Laplace operator with short range potential V (x), such thatWe study the equivalence of classical homogeneous Besov type spacesḂ s p (R), p ∈ (1, ∞) and the corresponding perturbed homogeneous Besov spaces associated with the perturbed Hamiltonian H = −∂ 2x + V (x) on the real line. It is shown that the assumptions 1/p < γ − 1 and zero is not a resonance guarantee that the perturbed and unperturbed homogeneous Besov norms of order s ∈ [0, 1/p) are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Besov spaces of order s ∈ [0, 1/p) invariant.