Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff–Neumann conditions hold at the internal vertices. Associated with this graph is a Schrödinger type operator L = −Δ + q(x) with Dirichlet boundary conditions at the two boundary nodes. Let {ω2n, φn(x)} be the eigenvalues and associated normalized eigenfunctions. Let v1 be a boundary vertex, and v2 the adjacent internal vertex. Assume we know the following data: {ω2nxφn(v1), φn(v2)}. Here, xφn(v2) refers to an outward normal derivative at v2 along one of the edges incident to the other internal vertex. From this data, we determine the following unknown quantities: the lengths of edges and the potential functions on each edge.
