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 aij, bj, and c, acting from the Sobolev space W1, 2(ℝn) to its dual W−1, 2(ℝn). This enables us to obtain analytic criteria for the fundamental notions of relative form boundedness, compactness, and infinitesimal form boundedness of ℒ︁ with respect to the Laplacian on L2(ℝn). 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′(ℝn). © 2005 Wiley Periodicals, Inc.