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

TL;DR

This paper investigates the equivalence of classical and perturbed homogeneous Besov spaces for 1D Schrödinger operators with short-range potentials, establishing conditions under which these spaces are norm-equivalent and invariant under wave operators.

Contribution

It provides new conditions ensuring the equivalence of Besov spaces for 1D Hamiltonians without zero resonance, extending understanding of function space invariance under perturbations.

Findings

Homogeneous Besov norms are equivalent under certain decay conditions on V(x).

Wave operators preserve classical homogeneous Besov spaces for specified s.

Zero resonance absence is crucial for norm equivalence.

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 $\dot{B}^s_p(R)$, $p \in (1,\infty)$ 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.