Unique continuation and absence of positive eigenvalues for Schrödinger operators. (With an appendix by E. M. Stein). (English) Zbl 0593.35119

The authors prove a strong unique continuation property: Let \(\Omega\) be a connected domain of \({\mathbb{R}}^ n\), \(x\in \Omega\), \(V\in L^{n/2}_{loc}({\mathbb{R}}^ n)\), \(q=2n/(n+2)\), \(u\in H^{2,q}_{loc}(\Omega)\), and \(u=0\) on some nonempty subset of \(\Omega\), then \(u=0\) on \(\Omega\). The method used is to prove a Carleman type inequality by complex interpolation. - The paper has an appendix by E. Stein where the proof is simplified and the result is extended to Lorentz spaces.
Reviewer: H.Siedentop


35R45 Partial differential inequalities and systems of partial differential inequalities
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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