  In this paper we study the number of bound states for potentials in one and
two spatial dimensions. We first show that in addition to the well-known fact
that an arbitrarily weak attractive potential has a bound state, it is easy to
construct examples where weak potentials have an infinite number of bound
states. These examples have potentials which decrease at infinity faster than
expected. Using somewhat stronger conditions, we derive explicit bounds on the
number of bound states in one dimension, using known results for the
three-dimensional zero angular momentum. A change of variables which allows us
to go from the one-dimensional case to that of two dimensions results in a
bound for the zero angular momentum case. Finally, we obtain a bound on the
total number of bound states in two dimensions, first for the radial case and
then, under stronger conditions, for the non-central case.
