  We use two renormalization techniques, Effective Field Theory and the
Similarity Renormalization Group, to solve simple Schr{\"o}dinger equations
with delta-function potentials in one and two dimensions. The familiar
one-dimensional delta-function does not require renormalization, but it
provides the simplest example of a local interaction that can be replaced by a
sequence of effective cutoff interactions that produce controllable power-law
errors. The two-dimensional delta-function leads to logarithmic divergences,
dimensional transmutation and asymptotic freedom, providing an example of some
of the most important renormalization problems in gauge field theories. We
concentrate on the power-law analysis of errors in low-energy observables. The
power-law suppression of the effects of irrelevant operators is critical to the
success of field theory, and understanding them turns renormalization group
techniques into powerful predictive tools for complicated problems where exact
solutions are not available.
