  We consider the Schr\"odinger equation for a relativistic point particle in
an external 1-dimensional $\delta$-function potential. Using dimensional
regularization, we investigate both bound and scattering states, and we obtain
results that are consistent with the abstract mathematical theory of
self-adjoint extensions of the pseudo-differential operator $H = \sqrt{p^2 +
m^2}$. Interestingly, this relatively simple system is asymptotically free. In
the massless limit, it undergoes dimensional transmutation and it possesses an
infra-red conformal fixed point. Thus it can be used to illustrate non-trivial
concepts of quantum field theory in the simpler framework of relativistic
quantum mechanics.
