  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.
