Quantitative unique continuation for Schrödinger operators

Abstract

We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for Δ+V. That is, for any non-trivial u that solves Δu+Vu=0 in some open, connected subset of Rn, we estimate the vanishing order of solutions in terms of the Lt-norm of V. Our results apply to all t>n2 and n≥3. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to Δu+Vu=0. To handle V∈Lt for every t∈(n2,∞], we prove a novel Lp−Lq Carleman estimate by interpolating a known Lp−L2 estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form Δu+W⋅∇u+Vu=0.