  These classical inequalities allow one to estimate the number of negative
eigenvalues and the sums $S_{\gamma}=\sum |\lambda_i|^{\gamma}$ for a wide
class of Schr\"{o}dinger operators. We provide a detailed proof of these
inequalities for operators on functions in metric spaces using the classical
Lieb approach based on the Kac-Feynman formula. The main goal of the paper is a
new set of examples which include perturbations of the Anderson operator,
operators on free, nilpotent and solvable groups, operators on quantum graphs,
Markov processes with independent increments. The study of the examples
requires an exact estimate of the kernel of the corresponding parabolic
semigroup on the diagonal. In some cases the kernel decays exponentially as
$t\to \infty $. This allows us to consider very slow decaying potentials and
obtain some results that are precise in the logarithmical scale.
