McLaughlin's inverse problem for the fourth-order differential operator

Abstract

In this paper, we revisit McLaughlin's inverse problem, which consists in the recovery of the fourth-order differential operator from the eigenvalues and two sequences of weight numbers. We for the first time prove the uniqueness for solution of this problem. Moreover, we obtain the interpretation of McLaughlin's problem in the framework of the general inverse problem theory by Yurko for differential operators of arbitrary orders. An advantage of our approach is that it requires neither smoothness of the coefficients nor self-adjointness of the operator. In addition, we establish the connection between McLaughlin's problem and Barcilon's three-spectra inverse problem.