Discrete analogy of the Gelfand-Levitan equation (Q2785323)

The origin of the method due to Gelfand and Levitan dates to early 50th. This method is one of those extensively used in applications. It provides the possibility of formulating necessary and sufficient conditions for existence in the large of the inverse problem solution. It is important in the view of nonlinearity of inverse problems. The essence of this method lies in reducing the nonlinear inverse problem (investigated originally in the spectral domain by Gelfand and Levitan) to a one-parameter set of linear Fredholm integral equations of the first or second kind. The authors consider the discrete analogy of a one-dimensional inverse problem for a hyperbolic equation. On the basis of the Gelfand-Levitan method a necessary and sufficient condition for the unique solvability of the discrete inverse problem is obtained.