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We consider the convergent problems of Dirichlet forms associated with theLaplace operators on thin domains. This problem appears in the field of quantumwaveguides. We study that a sequence of Dirichlet forms approximating theLaplace operators with the delta potential on thin domains Mosco converges tothe form associated with the Laplace operator with the delta potential on thegraph in the sense of Gromov-Hausdorff topology. From this results we can makeuse of many results established by Kuwae and Shioya about the convergence ofthe semigroups and resolvents generated by the infinitesimal generatorsassociated with the Dirichlet forms.

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