The Schrödinger and the relativistic Schrödinger operators on the energy space: boundedness and compactness criteria
by
Vladimir Maz'ya
Coauthors: I. Verbitsky

We give a complete characterization of the class of functions (or, more generally, complex-valued distributions) Q such that the following inequality holds:

| ó õ Rn  |u(x)|2 Q(x) dx| <= const ó õ Rn  |Ñu(x)|2dx,

where the constant is independent of u in C^\infty₀(Rⁿ). Similar inequalities are proved for the inhomogeneous Sobolev space W¹₂(Rⁿ). In other words, we establish a criterion for form-boundedness of Q relative to the Laplacian \Delta under no a priori assumptions on Q. For the Schrödinger operator L=-\Delta+Q, our criterion describes the class of admissible perturbations Q such that L:\ring L¹₂(Rⁿ) --> L ₂^-1(Rⁿ). We also establish similar boundedness and compactness criteria for the relativistic Schrödinger operator.

Date received: August 9, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahz-84.