DISCRETIZED LAPLACIANS ON AN INTERVAL AND THEIR RENORMALIZATION-GROUP

The Laplace operator admits infinite self-adjoint extensions when cons idered on a segment of the real line. They have different domains of e ssential self-adjointness characterized by a suitable set of boundary conditions on the wave functions. In this paper we show how these exte nsions can be recovered by studying the continuum limit of certain dis cretized versions of the Laplace operator on a lattice. Associated to this limiting procedure, there is a renormalization flow in the finite -dimensional parameter space describing the discretized operators. Thi s flow is shown to have infinite fixed points, corresponding to the se lf-adjoint extensions characterized by scale-invariant boundary condit ions. The other extensions are recovered by looking at the other traje ctories of the flow.