Point potential for the generator of a stable process

We consider a closed self-adjoint extension of the infinitesimal generator of a symmetric stable process whose domain core is Cc∞(Rd∖{0}). Such extensions of the Laplacian have been used in models of the hydrogen atom and more recently have modeled pinning at the origin of polymer models based on Brownian motion. We also outline the construction of pinned polymer models based on the symmetric stable processes, even when the underlying stable process does not possess a local time at the origin.