Solving Barcilon's inverse problems by the method of spectral mappings

In this paper, we consider Barcilon's inverse problem, which consists in the recovery of the fourth-order differential operator from three spectra. The relationship of Barcilon's three spectra with the Weyl-Yurko matrix is obtained. Moreover, the uniqueness theorem for the inverse problem solution is proved by developing the ideas of the method of spectral mappings. Our approach allows us to obtain the result for the general case of complex-valued distributional coefficients. In the future, the methods and the results of this paper can be generalized to differential operators of orders greater than 4 and used for further development of the inverse problem theory for higher-order differential operators.