On homogeneous Besov spaces for 1D Hamiltonians without zero resonance

We consider the 1-D Laplace operator with short-range potential V(x), such that (1+|x|)^γV(x)∈L¹(R),γ>1. We study the equivalence of classical homogeneous Besov type spaces B˙_p ^s(R), p∈(1,∞) and the corresponding perturbed homogeneous Besov spaces associated with the perturbed Hamiltonian H=−∂_x ²+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.