Inverse problems for quantum trees II:Recovering matching conditions for star graphs

Abstract

The inverse problem for the Schrödinger operator on a star graphis investigated. It is proven that such Schrödinger operator, i.e.the graph, the real potential on it andthe matching conditions at the central vertex, can be reconstructedfrom the Titchmarsh-Weyl matrix function associated with thegraph boundary. The reconstruction is also unique if the spectral datainclude not the whole Titchmarsh-Weyl function but its principal block(the matrix reduced by one dimension). The same result holds true ifinstead of the Titchmarsh-Weyl function the dynamical responseoperator or just its principal block is known.