Infinitesimal form boundedness and Trudinger's subordination for the Schrödinger operator

Abstract

We give explicit analytic criteria for two problems associated with the Schrödinger operator $H=-\Delta +Q$ on $L^2({\mathbb{R}}^n)$ where $Q\in D^{\prime }({\mathbb{R}}^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on $Q$ so that the quadratic form $<Q\cdot ,\cdot >$ has zero relative bound with respect to the Laplacian. For $Q\in L_{\mathrm{loc}}^1({\mathbb{R}}^n)$, this property can be expressed in the form of the integral inequality: $$|{\int }_{{\mathbb{R}}^n}|u(x){|}^{2}Q(x)dx|\le \epsilon ||\nabla u|{|}_{L^2({\mathbb{R}}^n)}^2+C(\epsilon )||u|{|}_{L^2({\mathbb{R}}^n)}^2,\forall u\in C_0^{\infty }({\mathbb{R}}^n),$$ for an arbitrarily small $\epsilon >0$ and some $C(\epsilon )>0$. Secondly, we characterize Trudinger's subordination property where $C(\epsilon )$ in the above inequality is subject to the condition $C(\epsilon )\le c{\epsilon }^{-\beta }$ ($\beta >0$) as $\epsilon \to +0$. Such quadratic form inequalities can be understood entirely in the framework of Morrey--Campanato spaces, using mean oscillations of $\nabla (1-\Delta {)}^{-1}Q$ and $(1-\Delta {)}^{-1}Q$ on balls or cubes. As a consequence, we characterize the class of those $Q$ which satisfy a multiplicative quadratic from inequality of Nash's type.

54 pages