On point interactions in two dimensions via additive functionals

Abstract

By taking the viewpoint of Brownian additive functionals, we extend an existing approximation theorem of the two-dimensional Laplacian singularly perturbed at the origin. The approximate operators are defined by adding a rescaled function to the two-dimensional Laplacian such that the support of the perturbation thus produced converges to a point in the limit of the rescaling, and suitable renormalizations of the coupling constants are also imposed. The first main theorem assumes the zero total mass of the function for rescaling. It proves the limits of the approximate resolvents up to an extended criticality defined by two levels of phase transitions. The second main theorem handles the case of positive total mass of the function for rescaling and proves an asymptotic expansion of a multivariate density induced by the approximate resolvent. Moreover, this expansion is valid beyond criticality.