Derivation of optimal LAP estimates for perturbed 2D Laplacian via rank 1 perturbations and properties of virtual states.

We recall that virtual levels are the points of the essential spectrum at whose vicinity the limiting absorption principle (in particular spaces) breaks down. We describe an elementary construction which allows one to derive optimal LAP estimates for 1D and 2D Laplace operator perturbed so that the virtual level at zero is removed. We also show — also for the Laplacian in 1D and 2D — that if a perturbation is such that there is a virtual level at the origin, then the corresponding virtual state is not necessarily from $L^\infty$.