Unique continuation for Schrödinger operators in dimension three or less (Q791024)

We show that the differential inequality \(| \Delta u| \leq v| u|\) has the unique continuation property relative to the Sobolev space \(H^{2,1}\!\!\!_{loc}(\Omega)\), \(\Omega \subset R^ n\), \(n\leq 3\), if v satisfies the condition \[ (K_ n\!^{loc})\quad \lim_{r\to 0}\sup_{x\in K}\int_{| x-y|<r}| x-y|^{2-n}v(y)dy=0 \] for all compact \(K\subset \Omega\), where if \(n=2\), we replace \(| x-y|^{2-n}\) by -lo\(g| x-y|\). This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, \(H=-\Delta +v\), in the case \(n\leq 3\). The proof uses Carleman's approach together with the following pointwise inequality valid for all \(N=0,1,2,..\). and any \(u\in H_ c\!^{2,1}(R^ 3-\{0\});\) \[ \frac{| u(x)|}{| x|^ N}\leq C\int_{R^ 3}| x-y|^{- 1}\frac{| \Delta u(y)|}{| y|^ N}dy \] for a.e. x in \(R^ 3\).