  We provide a class of necessary and sufficient conditions for the
discreteness of spectrum of Schr\"odinger operators with scalar potentials
which are semibounded below. The classical discreteness of spectrum criterion
by A.M.Molchanov (1953) uses a notion of negligible set in a cube as a set
whose Wiener's capacity is less than a small constant times the capacity of the
cube. We prove that this constant can be taken arbitrarily between 0 and 1.
This solves a problem formulated by I.M.Gelfand in 1953. Moreover, we extend
the notion of negligibility by allowing the constant to depend on the size of
the cube. We give a complete description of all negligibility conditions of
this kind. The a priori equivalence of our conditions involving different
negligibility classes is a non-trivial property of the capacity. We also
establish similar strict positivity criteria for the Schr\"odinger operators
with non-negative potentials.
