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

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

2004-06-02

arxiv.org/abs/math/0406050v1

Invent. Math. 162 (2005), 81 - 136

Mathematics

Functional Analysis

54 pages

Scientific paper

10.1007/s00222-005-0439-y

We give explicit analytic criteria for two problems associated with the Schr\"odinger operator $H = -\Delta + Q$ on $L^2(\R^n)$ where $Q\in D'(\R^n)$ is an arbitrary real- or complex-valued potential. First, we obtain necessary and sufficient conditions on $Q$ so that the quadratic form $$ has zero relative bound with respect to the Laplacian. For $Q\in L^1_{\rm loc}(\R^n)$, this property can be expressed in the form of the integral inequality: $$ | \int_{\R^n} |u(x)|^2 Q(x) dx | \leq \epsilon ||\nabla u||^2_{L^2(\R^n)} + C(\epsilon) ||u||^2_{L^2(\R^n)}, \quad \forall u \in C^\infty_0(\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.

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