On transparent vertex boundary conditions for quantum graphs

Absence of reflection is important for many fields of physics and engineering particularly, for nanoelectronics. There are a few ways to reach such property. One of them is related to ensuring of proper boundary condition. Ballistic electron transport in quite a few nanosystems can be modeled by quantum graphs. For quantum graphs, absence of reflection is related to the transparency of the graph vertices. In the present paper, the scattering problem for star-like quantum graph is studied. The free Schrodinger operator is considered at each edge of the graph. We derive the boundary conditions at the graph vertices (i.e., coupling conditions) which ensure low reflection. The technique is based on rational function Pade approximation for the dispersion relation, i.e., the suggested condition is not exact but an approximate one. Two computational experiments with transmission of different wave packets show low reflection of the electron.