A Limit Evolution Problem for Time-Dependent Point Interactions

We study the diffusion inR3of a particle interacting withNfixed points through point interactions whose strength varies in time. Under mild assumptions on the time dependence of the strengths, we prove existence for all times and uniqueness of the solution, for which we provide a rather explicit expression. We also prove that, under a suitable rescaling of the interaction strengths, the solution converges, whenN→∞, to the solution of a diffusion equation with a regular killing term (potential). We use properties of the local self-adjoint extensions of the Laplacian and results from the theory of fractional integrals and derivatives.