Form boundedness of the general second‐order differential Operator

We give explicit necessary and sufficient conditions for the boundedness of the general second‐order differential operator$${\cal L} = \sum\limits_{i,\,j=1}^{n} a_{ij} \partial_{i} \partial_{j} + \sum\limits_{j=1}^{n} b_{j} \partial_{j} + c$$ with real‐ or complex‐valued distributional coefficients a ij , b j , and c , acting from the Sobolev space W 1, 2 (ℝ n ) to its dual W −1, 2 (ℝ n ). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of ℒ with respect to the Laplacian on L 2 (ℝ n ). In particular, we establish a complete characterization of the form boundedness of the Schrödinger operator $(i \nabla + \vec{a})^2 + q$ with magnetic vector potential $\vec{a} \in L^2_{{\rm loc}} (R^{n})^{n}$ and q ∈ D′ (ℝ n ). © 2005 Wiley Periodicals, Inc.