On homogeneous Besov spaces for $1D$ Hamiltonians without zero resonance

Abstract

We consider 1-D Laplace operator with short range potential V(x), such that $$(1+|x|{)}^{\gamma }V(x)\in L^1(R),\gamma >1.$$ We study the equivalence of classical homogeneous Besov type spaces ${\dot{B}}_p^s(R)$, $p\in (1,\infty )$ and the corresponding perturbed homogeneous Besov spaces associated with the perturbed Hamiltonian $\mathcal{H}=-{\partial }_x^2+V(x)$ on the real line. It is shown that the assumptions $1/p<\gamma -1$ and zero is not a resonance guarantee that the perturbed and unperturbed homogeneous Besov norms of order $s\in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Besov spaces of order $s\in [0,1/p)$ invariant.