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

TL;DR

This paper investigates the equivalence of Sobolev spaces associated with a perturbed Hamiltonian on the real line, establishing conditions under which the classical and perturbed Sobolev norms are equivalent, with implications for wave operator invariance.

Contribution

It demonstrates that zero not being a resonance ensures the equivalence of Sobolev norms for perturbed and unperturbed Hamiltonians, extending the understanding of Sobolev space invariance under perturbations.

Findings

Sobolev norm equivalence under zero resonance condition

Invariance of Sobolev spaces under wave operators

Extension of classical Sobolev space theory to perturbed Hamiltonians

Abstract

We consider the following perturbed Hamiltonian $\mathcal{H}= -\partial_x^2 + V(x)$ on the real line. The potential $V(x)$ is a real - valued function of short range type. We study the equivalence of classical homogeneous Sobolev type spaces $\dot{H}^s_p$, $p \in (1,\infty)$ 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 $s = \gamma - 1 \in [0,1/p)$ are equivalent. As a corollary, the corresponding wave operators leave classical homogeneous Sobolev spaces of order $s \in [0,1/p)$ invariant.