The form boundedness criterion for the relativistic Schr\"odinger operator

Abstract

We establish necessary and sufficient conditions for the boundedness of the relativistic Schr\"odinger operator $\mathcal{H} = \sqrt{-\Delta} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary real- or complex-valued potential $Q$ on $\R^n$. %Analogous results for %$\mathcal{H}_m = \sqrt{-\Delta + m^2} - m + Q$, as well as %the corresponding compactness criteria are obtained. In other words, we give a complete solution to the problem of the domination of the potential energy by the kinetic energy in the relativistic case characterized by the inequality $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \text{const} ||u||^2_{W_2^{1/2}}, \quad u \in C^\infty_0(\R^n), $$ where the ``indefinite weight'' $Q$ is a locally integrable function (or, more generally, a distribution) on $\R^n$. Along with necessary and sufficient results, we also present new broad classes of admissible potentials $Q$ in the scale of Morrey spaces of negative order, and discuss their relationship to well-known $L_p$ and Fefferman-Phong conditions.