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We discuss approximations of the vertex coupling on a star-shaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the CheonShigehara technique using interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the {n+1\choose 2}-parameter family of all time-reversal invariant couplings.

We discuss discuss spectral and scattering properties of a particle confined to a straight Dirichlet tube in R''with a family of point interactions.

We study approximations of billiard systems by lattice graphs. It is demonstrated that under natural assumptions the graph wavefunctions approximate solutions of the Schroedinger equation with energy rescaled by the billiard dimension. As an example, we analyze a Sinai billiard with attached leads. The results illustrate emergence of global structures in large quantum graphs and offer interesting comparisons with patterns observed in complex networks of a different nature.

We adopt a recent work in Chung, Qian, Uhlmann and Zhao (Inverse Problems, 23 (2007) 309-329) to develop a phase space method for reconstructing pressure wave speed and shear wave speed of an elastic medium from travel time measurements. The method is based on the so-called Stefanov-Uhlmann identity which links two Riemannian metrics with their travel time information. We design a numerical algorithm to solve the resulting inverse problem. The algorithm is a hybrid approach that combines both Lagrangian and Eulerian formulations. In particular the Lagrangian formulation in phase space can take into account multiple arrival times naturally, while the Eulerian formulation for wave speeds allows us to compute the solution in physical space. Numerical examples are shown to validate the method.