On the solvability of inverse transmission eigen-values problem for wave equation with a spherically-symmetric piecewise-constant wave speed

The inverse spectral problem is examined in the paper for a spherically-symmetric piecewise-constant media, where transmission eigen-values are given as initial data. The problem is reduced to Hilbert-Riemann problems and a condition of uniqueness of the problem solution is established in a form, which is less strict in comparison with the inverse spectral problem for continuous wave speed. A constructive procedure for reconstruction of piecewise-constant wave speed is proposed starting form for all given complex transmission eigen-values including theirs multiplicities.