Approximations of singular vertex couplings in quantum graphs

Abstract

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 $\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)$-parameter family of all time-reversal invariant couplings.

LaTeX source file, 33 pages, with 3 eps figures

References in corpus (2)

• Scattering solutions in a network of thin fibers: small diameter asymptotics
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