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We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.

In this paper, we develop an adaptive Generalized Multiscale Discontinuous Galerkin Method (GMs-DGM) for a class of high-contrast flow problems, and derive a-priori and a-posteriori error estimates for the method. Based on the a-posteriori error estimator, we develop an adaptive enrichment algorithm for our GMsDGM and prove its convergence. The adaptive enrichment algorithm gives an automatic way to enrich the approximation space in regions where the solution requires more basis functions, which are shown to perform well compared with a uniform enrichment. We also discuss an approach that adaptively selects multiscale basis functions by correlating the residual to multiscale basis functions (cf.[4]). The proposed error indicators are L2-based and can be inexpensively computed which makes our approach efficient. Numerical results are presented that demonstrate the robustness of the proposed error indicators.