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Approximations of singular vertex couplings in quantum graphs

Pavel Exner·Ondrej Turek
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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 ${n+1\choose 2}$-parameter family of all time-reversal invariant couplings.

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Pavel Exner, Ondrej Turek. 2007-03-16. Approximations of singular vertex couplings in quantum graphs. https://doi.org/10.1142/s0129055x07003073

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