Abstract
We consider a selfadjoint operator,A, and a selfadjoint rank-one projection,P, onto a vector, φ, which is cyclic forA. We study the set of all eigenvalues of the operatorA t =A+tP (t∈∝) that belong to its essential spectrum (which does not depend on the parametert). We prove that this set is empty for a dense set of values oft. Then we apply this result or its idea to questions of Anderson localization for 1-dimensional Schrödinger operators (discrete and continuous).
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