The Mosco convergence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains

TL;DR

This paper proves that Dirichlet forms approximating Laplace operators with delta potentials on thin domains Mosco converge to a form on a graph, enabling the transfer of convergence results for semigroups and resolvents.

Contribution

It establishes the Mosco convergence of Dirichlet forms for Laplace operators with delta potentials on thin domains to a graph-based form, linking quantum waveguides to graph models.

Findings

Mosco convergence of Dirichlet forms on thin domains

Convergence to Laplace operator with delta potential on a graph

Application of Kuwae and Shioya's results on semigroup convergence

Abstract

We consider the convergent problems of Dirichlet forms associated with the Laplace operators on thin domains. This problem appears in the field of quantum waveguides. We study that a sequence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains Mosco converges to the form associated with the Laplace operator with the delta potential on the graph in the sense of Gromov-Hausdorff topology. From this results we can make use of many results established by Kuwae and Shioya about the convergence of the semigroups and resolvents generated by the infinitesimal generators associated with the Dirichlet forms.