The form boundedness criterion for the relativistic Schrödinger operator

Abstract

We establish necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator $\mathcal{H}=\sqrt{-\Delta }+Q$ from the Sobolev space $W_2^{1/2}({\mathbb{R}}^n)$ to its dual $W_2^{-1/2}({\mathbb{R}}^n)$, for an arbitrary real- or complex-valued potential $Q$ on ${\mathbb{R}}^n$. %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 $$|{\int }_{{\mathbb{R}}^n}|u(x){|}^{2}Q(x)dx|\le \text{const}||u|{|}_{W_2^{1/2}}^2,u\in C_0^{\infty }({\mathbb{R}}^n),$$ where the ``indefinite weight'' $Q$ is a locally integrable function (or, more generally, a distribution) on ${\mathbb{R}}^n$. 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.

to appear in Ann. Inst. Fourier (Grenoble)