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

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.