Kreĭn formula and convergence of hamiltonians with scaled potentials in dimension one

In this brief report we study the convergence of the Hamiltonian hε := −(⋅)′′ + V (x∕ε)∕ε2 in dimension one as ε goes to zero. This problem has already been studied in several former works (also in the more general setting of metric graphs) and the results that we present here are not new. Aim of this work is to formulate the problem in the setting of metric graphs and to exploit an approach based on a Kreı̆n formula for the resolvent of hε. Such a formula allows to mark out the rôle of the zero eigenvalue for an auxiliary Hamiltonian. The existence of the zero eigenvalue is responsible of the coupling in the limiting Hamiltonian, otherwise hε converges in norm resolvent sense to the direct sum of two Dirichlet Laplacians on the half-line. In a forthcoming paper such approach will be generalized to the study of an analogous problem on metric graphs with a small compact core.