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

TL;DR

This paper characterizes the conditions under which the relativistic Schrödinger operator with potential Q is bounded between specific Sobolev spaces, providing a complete solution to potential energy domination in the relativistic setting.

Contribution

It establishes necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator with arbitrary potentials, including new classes of admissible potentials in Morrey spaces.

Findings

Derived complete boundedness criteria for the operator.

Identified broad classes of potentials in Morrey spaces.

Connected results to classical $L_p$ and Fefferman-Phong conditions.

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.