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The occurrence of complex potentials with real eigenvalues has implications concerning the inverse problem, i.e. the determination of a potential from its spectrum. First, any complex potential with real eigenvalues has at least one equivalent local potential. Secondly, a real spectrum does not necessarily corresponds to a local real potential. A basic ambiguity arises from the possibility the spectrum to be generated by a complex potential. The purpose of this work is to discuss several aspects of this problem.

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We extend to complex potentials a method developed to solve the inverse problem from bound states in the case of a local real potential. A first example is presented, which is based on a complex version of the Kratzer potential. In this case, the Schrödinger equation admits analytical solutions, providing us with a test of the method. The application to the π − -28 Si and K − -208 Pb hadronic atoms shows the possibilities and limitations of our approach.