  We study a self-interacting scalar field theory in the presence of a
\delta-function background potential. The role of surface interactions in
obtaining a renormalizable theory is stressed and demonstrated by a two-loop
calculation. The necessary counterterms are evaluated by adopting dimensional
regularization and the background field method. We also calculate the effective
potential for a complex scalar field in a non-simply connected spacetime in the
presence of a \delta-function potential. The effective potential is evaluated
as a function of an arbitrary phase factor associated with the choice of
boundary conditions in the non-simply connected spacetime. We obtain asymptotic
expansions of the results for both large and small \delta-function strengths,
and stress how the non-analytic nature of the small strength result vitiates
any analysis based on standard weak field perturbation theory.
