Semiclassical estimates for the magnetic Schr\"odinger operator on the line

TL;DR

This paper establishes weighted Carleman estimates for one-dimensional magnetic Schr"odinger operators with low regularity potentials, leading to optimal resolvent estimates and applications to resonance distribution and evolution equations.

Contribution

It introduces a novel Carleman estimate for 1D Schr"odinger operators with weak regularity assumptions on potentials, extending previous methods to less regular settings.

Findings

Proves weighted Carleman estimates under minimal regularity conditions.

Derives optimal limiting absorption resolvent estimates.

Applies results to resonance distribution and evolution equations.

Abstract

We prove a weighted Carleman estimate for a class of one-dimensional, self-adjoint Schr\"odinger operators $P(h)$ with low regularity electric and magnetic potentials, where $h > 0$ is a semiclassical parameter. The long range part of either potential has bounded variation. The short range part of the magnetic potential belongs to $L^1(\mathbb{R}) \cap L^2(\mathbb{R})$, while the short range part of the electric potential is a finite signed measure. The proof is a one dimensional instance of the energy method, which is used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result lies in the weak regularity assumptions on the coefficients. As a consequence of the Carleman estimate, we establish an optimal limiting absorption resolvent estimates for $P(h)$. We also present standard applications to the distribution of resonances for $P(1)$ and to associated evolution equations.