Virtual levels and virtual states of linear operators in Banach spaces. Applications to Schroedinger operators

Virtual levels, also known as threshold resonances, admit several equivalent characterizations: (1) there are corresponding virtual states from a space slightly weaker than L; (2) there is no limiting absorption principle in their vicinity (e.g. no weights such that the “sandwiched” resolvent is uniformly bounded); (3) an arbitrarily small perturbation can produce an eigenvalue. We develop a general approach to virtual levels in Banach spaces and provide applications to Schrödinger operators with nonselfadjoint potentials and in any dimension.

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