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In this paper, we consider Barcilon's inverse problem, which consists of therecovery of the fourth-order differential operator from three spectra. Weobtain the relationship of Barcilon's three spectra with the Weyl-Yurko matrix.Moreover, we prove the uniqueness theorem for the inverse problem solution bydeveloping the ideas of the method of spectral mappings. Our approach allows usto obtain the result for the general case of complex-valued distributionalcoefficients. In the future, the methods and the results of this paper can begeneralized to differential operators of orders greater than 4 and used forfurther development of the inverse problem theory for higher-order differentialoperators.

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