Resolvent estimates for the Schr\"odinger operator with $L^\infty$ electric and magnetic potentials and applications to the local energy decay

Abstract

We establish resolvent estimates that extend earlier results to a larger class of electric potentials $V\in L^\infty(\mathbb{R}^d;\mathbb{R})$, $d\ge 3$, and magnetic potentials $b\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)$ such that $V(x), b(x)=O_k\left(|x|^{-k}\right)$, $|x|\gg 1$, for every integer $k$. More precisely, we prove estimates for the derivatives of the weighted resolvent of the corresponding magnetic Schr\"odinger operator, which are uniform with respect to both the spectral parameter and the order of derivation. We also show that these resolvent estimates still hold for the Dirichlet self-adjoint realization of the Schr\"odinger operator in the exterior of a non-trapping obstacle in $\mathbb{R}^d$, $d\ge 2$, provided the magnetic potential is supposed identically zero. As an application of these resolvent estimates, we obtain the rate of decay of the local energy of solutions to the corresponding wave equation. In particular, we show that for potentials satisfying $|V(x)|+|b(x)|\le Ce^{-c|x|^s}$, $c,C>0$, $0<s<1$, the rate of decay of the local energy is $e^{-c_0t^s}$ with some constant $c_0>0$, where $t\gg 1$ is the time variable.