arXiv math-ph/03090312003-09-12math-phmath.FAmath.MP

相对论性薛定谔算子的形式有界性判据

The form boundedness criterion for the relativistic Schrödinger operator

V. G. Maz'ya, I. E. Verbitsky

备注：: to appear in Ann. Inst. Fourier (Grenoble)

摘要：

对于$\R^n$上的任意实值或复值位势$Q$，我们建立了相对论性薛定谔算子$\mathcal{H} = \sqrt{-Δ} + Q$从Sobolev空间$W^{1/2}_2 (\R^n)$到它的对偶$W^{-1/2}_2 (\R^n)$有界的充要条件.得到了% $\mathcal{H}_m = \sqrt{-Δ+ m^2} - m + Q$的类似结果以及相应的紧性准则.换句话说，我们给出了一个完整的解决方案，在相对论的情况下，其特征在于由不等式$$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \text{const} ||u||^2_{W_2^{1/2}}, \quad u \in C^\infty_0(\R^n), $$的“不定权重”$Q$是一个局部可积函数（或更一般地说，分布）上$\R^n$的动能的势能支配的问题。除了充分必要的结果外，我们还在负序Morrey空间的尺度上给出了新的广义容许势$Q$，并讨论了它们与著名的$L_p$和Festhiman-Phong条件的关系.

英文摘要：

We establish necessary and sufficient conditions for the boundedness of the relativistic Schrödinger operator $\mathcal{H} = \sqrt{-Δ} + Q$ 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 %$\mathcal{H}_m = \sqrt{-Δ+ m^2} - m + Q$, 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.

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