  Simon's results on the negative spectrum of recurrent Schr\"{o}dinger
operators ($d=1,2$) are extended to a wider class of potentials and to
non-local operators. An example of $L^1-$potental is constructed for which the
essential spectrum of two-dimensional Schr\"{o}dinger operator covers the whole
axis. Some counterexamples are provided for transient operators ($d\geq3$)
showing that the assumptions on the potential for the validity of the
Cwikel-Lieb-Rozenblum estimate can't be improved significantly.
