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

Abstract

We develop a general approach to virtual levels in Banach spaces. We show that virtual levels admit several characterizations which are essentially equivalent: (1) there are corresponding virtual states (from a certain larger space); (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 provide applications to Schr\"odinger operators with nonselfadjoint nonlocal potentials and in any dimension, deriving resolvent estimates in the neighborhood of the threshold when the corresponding operator has no virtual level there.