Virtual levels and virtual states of operators in Banach spaces.

Abstract: Virtual levels admit several equivalent characterizations:
(1) there are corresponding eigenstates from L^{^2} or a space “slightly weaker” than L^{^2};
(2) there is no limiting absorption principle in the vicinity of a virtual level (e.g. no weights such that the “sandwiched” resolvent remains 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 Schroedinger operators with non selfadjoint potentials and in any dimension.