Green functions on product networks. (English) Zbl 1418.90048
Summary: We aim here at determining the Green function for general Schrödinger operators on product networks. The first step consists in expressing Schrödinger operators on a product network as sum of appropriate Schrödinger operators on each factor network. Hence, we apply the philosophy of the separation of variables method in PDE, to express the Green function for the Schrödinger operator on a product network using Green functions on one of the factors and the eigenvalues and eigenfunctions of some Schrödinger operator on the other factor network. We emphasize that our method only needs the knowledge of eigenvalues and eigenfunctions of one of the factors, whereas other previous works need the spectral information of both factors. We apply our results to compute the Green function of \(P_m \times S_h\), where \(P_m\) is a Path with \(m\) vertices and \(S_h\) is a Star network with \(h + 1\) vertices.
MSC:
| 90B10 | Deterministic network models in operations research |
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 31C20 | Discrete potential theory |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
Keywords:
discrete potential theory; Schrödinger operators; Green functions; product networks; spectra; separation of variablesReferences:
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