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 δ 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 2´-parameter family of all time-reversal invariant couplings.

Quantum Waveguides

“…An extension to vertices of higher degrees was done for the δ s -coupling in [CE04]; Theorem 8.2 is taken from this paper. The method based on adding δ vertices also works in cases without an edge-permutation symmetry giving at most a 2n parameter family of vertex couplings [ET07].…”

Vertex Couplings in Quantum Graphs: Approximations by Scaled Schrödinger Operators

“…The idea put forward in Ref. 21 was to change locally the graph topology by adding new edges in the vicinity of the vertex whose lengths shrink to zero in the approximation. This yielded a family of couplings with n+1 2 parameters and real matrices A, B.…”

Approximation of a general singular vertex coupling in quantum graphs

“…We see from (6) that m is equal to the rank of the matrix applied at Ψ V . We observe that the rank of this matrix, as well as of that applied at Ψ V , is not influences by any of the manipulations mentioned above, hence it is obvious that m = rank(B) and that such a choice is the only possible, i.e.…”