On the Reconstruction of the Boundary Conditions for Star Graphs Sergei Avdonin, Pavel Kurasov, and Marlena Nowaczyk Abstract. The Laplace operator on a star graph is considered. The prob- lem to recover the vertex matching boundary conditions from a part of the scattering matrix is investigated. 1. Introduction Differential operator on geometric graphs have been studied from the beginning of 80-ies [8, 11], but recent interest in nano-structures has led to enormous interest in mathematical studies of the problem [14, 16, 17, 19]. In this article we discuss the possibility to reconstruct the matching (boundary) conditions at the unique vertex of a star graph from the corresponding scattering matrix. This problem can easily be solved if the total scattering matrix is known (see [15]), and it has been shown recently that the scattering matrix at a particular value of the energy can effectively be used to uniquely parameterize the matching conditions [18]. The problem we are interested in is the possibility to reconstruct the matching conditions if only a part of the scattering matrix is known, more precisely the principal (v - 1) × (v - 1) block (S v (k 0 )) v;v , where v is the valency of the vertex. This problem can be considered as the first step towards reconstruction of the vertex matching conditions for trees from the corresponding scattering matrix. The problem of reconstructing the Schr¨ odinger operator on a star graph was first discussed by N.I. Gerasimenko and B.S. Pavlov [11, 12] using the Gelfand- Levitan-Marchenko method. The inverse spectral and scattering problems for trees have intensively been studied in recent years by M. Belishev, M. Brown, R. Carlson, G. Freiling, A. Vakulenko, R. Weikard, V. Yurko, and the authors [1, 2, 4, 5, 6, 7, 9, 10, 21]. It has been proven that the knowledge of the Dirichlet-to-Neumann map, or Titchmarsh-Weyl matrix function allows one to calculate the potential for standard boundary conditions at the vertices. The case of more general boundary conditions has been discussed in [10], but the whole family of boundary conditions has not been investigated yet. 1991 Mathematics Subject Classification. Primary 81C05, 35R30, 35L05, 93B05, 49E15. Key words and phrases. quantum graphs, inverse problems, matching conditions. S.A.’s research is supported in part by the National Science Foundation, grants ARC 0724860 and DMS 0648786; P.K.’s research is supported in part by the grants from Swedish Research Coun- cil and The Swedish Royal Academy of Sciences. The authors would like to thank V.Ufnarovski for helpful discussions. 1