An approximation to    couplings on graphs

Abstract

We discuss a general parametrization for vertices of quantum graphs and show, in particular, how the δ ′ s and δ ′ coupling at an n edge vertex can be approximated by means of n + 1 couplings of the δ type provided the latter are properly scaled.

A remark on Krein's resolvent formula and boundary conditions

“…For the use of this formula in the traditional way one needs a kind of preliminary construction, like finding a maximal common part of two extensions, see [2,Appendix A]. While this is enough for many applications, including models with point interactions, there is a number of problems like the study of quantum graphs or more general hybrid structures, where self-adjoint extensions are suitable described by more complicated boundary conditions, see [5,6,7], and it is necessary to modify Krein's resolvent formula to take into account these new needs. This can be done if one either modifies the coordinates in which the boundary data are calculated [6,8] or considers boundary conditions given in a non-operator way using linear relations [9,10].…”

Reconstruction of the Sturm-Liouville operators on a graph with δ′ s couplings

“…A graph with δ ′ s couplings was introduced and investigated by Peter Exner, and Peter Kuchment, and so on. A graph with δ ′ s couplings has important applications in lattice Kronig-Penney models, and the δ ′ s couplings at a d adge vertex can be approximated by means of d +1 couplings of the δ-type [9]. The question of physical meaning of such a coupling on graphs was addressed and a pair of simple nontrivial examples of the so-called δ ′ s couplings was presented in [12,13].…”

Leaky quantum graphs: A review

“…2 An alternative approach is to keep the graph fixed and to approximate the vertex coupling through suitably scaled families of regular or singular interactions -see [Ex96,CE04,ETu07].…”