  We examine several zero-range potentials in non-relativistic quantum
mechanics. The study of such potentials requires regularization and
renormalization. We contrast physical results obtained using dimensional
regularization and cutoff schemes and show explicitly that in certain cases
dimensional regularization fails to reproduce the results obtained using cutoff
regularization. First we consider a delta-function potential in arbitrary space
dimensions. Using cutoff regularization we show that for $d \ge 4$ the
renormalized scattering amplitude is trivial. In contrast, dimensional
regularization can yield a nontrivial scattering amplitude for odd dimensions
greater than or equal to five. We also consider a potential consisting of a
delta function plus the derivative-squared of a delta function in three
dimensions. We show that the renormalized scattering amplitudes obtained using
the two regularization schemes are different. Moreover we find that in the
cutoff-regulated calculation the effective range is necessarily negative in the
limit that the cutoff is taken to infinity. In contrast, in dimensional
regularization the effective range is unconstrained. We discuss how these
discrepancies arise from the dimensional regularization prescription that all
power-law divergences vanish. We argue that these results demonstrate that
dimensional regularization can fail in a non-perturbative setting.
