Counterexample to Barcilon’s Uniqueness Theorem for the Fourth-Order Inverse Spectral Problem

In this paper, we construct a counterexample to the uniqueness theorem by Barcilon (Geophys J Int 38(2):287–298, 1974), which is well-known in the field of inverse spectral problems. Our example shows that Barcilon’s three spectra do not uniquely specify the coefficients of the fourth-order differential equation. Our technique is based on the method of spectral mappings, which is a universal tool in the inverse spectral theory for higher-order differential operators. The example is obtained by a finite perturbation of the spectral data for the trivial problem with the zero coefficients. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.