Abstract
Consider a three-edge star graph, made up of unknown Sturm-Liouville operators on each edge. By using the heat propagation through the graph and measuring the heat transfer occurring at its vertices, we show that we can extract enough spectral data to reconstruct the three Sturm-Liouville operators by using the Gelfand-Levitan theory. Furthermore this reconstruction is achieved by a single measurement provided we use a special initial condition.
Highlights
Consider the heat equation on a simple star graph with three equal edges∂tu(j)(x, t) = ∂x2u(j)(x, t) − q(j)(x)u(j)(x, t) for 0 < x < a, t > 0, u(1)(0, t) = u(2)(0, t) = u(3)(0, t) = 0, (a) (1)u(1)(a, t) = u(2)(a, t) = u(3)(a, t), (b)∂xu(1)(a, t) + ∂xu(2)(a, t) + ∂xu(3)(a, t) = 0,(c) u(j) (x, 0) = f (j)(x) for j = 1, 2, 3.Equation (1) describes the heat propagation along the three edges of a star graph with unknown coefficients q(j)(x) ≤ 0
Condition (1b) describes the fact that they have the same temperature at the common vertex x = a while Kirchoff’s law, condition (1c), says that the heat transferred through the node x = a is conserved
The inverse problem is to recover the coefficient or potential q = q(1), q(2), q(3) T from readings of the temperature and heat transfer at the vertices of the graph defined by the map, T
Summary
Since they should differ at least on one edge, say jth , φ(nj) = φn(j), it follows that φ(nj) − φ(nj) is a nontrivial eigensolution of the Sturm-Liouville problem with the Dirichlet boundary conditions on the jth-edge, and so λn ∈ σD(j) for one j ∈ {1, 2, 3}, which is again impossible. We use the observed coefficients to find out about the type of the extracted eigenvalues We have the following cases: A1) λn is a simple eigenvalue of type (I) if and only if A(nj) = 0, j = 1, 2, 3.
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