  In this paper we argue that boundary condition may run with energy scale. As
an illustrative example, we consider one-dimensional quantum mechanics for a
spinless particle that freely propagates in the bulk yet interacts only at the
origin. In this setting we find the renormalization group flow of U(2) family
of boundary conditions exactly. We show that the well-known scale-independent
subfamily of boundary conditions are realized as fixed points. We also discuss
the duality between two distinct boundary conditions from the renormalization
group point of view. Generalizations to conformal mechanics and quantum graph
are also discussed.
