An inverse problem on a metric graph with cycle
Abstract
Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schr\"odinger type operator L=- +q(x) with Dirichlet boundary conditions at the two boundary nodes. Let \ ωn2, \ n(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: \ ωn2,∂x n(v1),∂xn(v2)\. Here ∂xn(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.
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