Abstract
Let L = −Δ− W be a Schrödinger operator with a potential \(W\in L^{\frac{n+1}{2}}(\mathbb{R}^n)\), \(n \geq 2\). We prove that there is no positive eigenvalue. The main tool is an \(L^{p}-L^{p^\prime}\) Carleman type estimate, which implies that eigenfunctions to positive eigenvalues must be compactly supported. The Carleman estimate builds on delicate dispersive estimates established in [7]. We also consider extensions of the result to variable coefficient operators with long range and short range potentials and gradient potentials.
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Communicated by B. Simon
The first author was partially supported by DFG grant KO1307/1 and also by MSRI for Fall 2005
The second author was partially supported by NSF grants DMS0354539 and DMS 0301122 and also by MSRI for Fall 2005
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Koch, H., Tataru, D. Carleman Estimates and Absence of Embedded Eigenvalues. Commun. Math. Phys. 267, 419–449 (2006). https://doi.org/10.1007/s00220-006-0060-y
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DOI: https://doi.org/10.1007/s00220-006-0060-y