Spectral Inequalities for the Schr{ö}dinger operator

Abstract

In this paper we deal with the so-called "spectral inequalities", which yield a sharp quantification of the unique continuation for the spectral family associated with the Schrödinger operator in ${\mathbb{R}}^d$ \begin{equation*} H_{g,V} = Δ_g + V(x), \end{equation*} where ${\Delta }_g$ is the Laplace-Beltrami operator with respect to an analytic metric $g$, which is a perturbation of the Euclidean metric, and $V(x)$ a real valued analytic potential vanishing at infinity.

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• Propagation of smallness and spectral estimates
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• Quantitative unique continuation and observability on an equidistributed set for the diffusion equation in R^N