An inverse problem for multidimensional first-order systems

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Abstract
An inverse problem associated with N first-order equations in n+1 dimensions, n>1, is considered: Given appropriate inverse data T reconstruct the potential q(x0,x), where q is an N×N off-diagonal matrix. Although q depends on n+1 variables, it turns out that T depends on 3n−1 variables. This necessitates imposing certain constraints on T, i.e., T must be suitably characterized. The characterization problem for T is solved explicitly. Furthermore, the problem of reconstructing q is reduced to one for reconstructing a 2×2 matrix potential in two dimensions. The inverse data needed for the reduced problem are obtained in closed form from T. A method for solving two-dimensional inverse problems has recently appeared in the literature.

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