Approximations of singular vertex couplings in quantum graphs

Details Approximations of singular vertex couplings in quantum graphs Approximations of singular vertex couplings in quantum graphs

2007-03-16

arxiv.org/abs/math-ph/0703051v1

Physics

Mathematical Physics

LaTeX source file, 33 pages, with 3 eps figures

Scientific paper

10.1142/S0129055X07003073

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 Cheon-Shigehara technique using $\delta$ 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.

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