McLaughlin's Inverse Problem for the Fourth-Order Differential Operator

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 norming constants. We prove the uniqueness for solution of this problem for the first time. Moreover, we obtain an 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 the smoothness of the coefficients nor the self-adjointness of the operator. In addition, we establish the connection between McLaughlin's problem and Barcilon's three-spectra inverse problem. DOI 10.1134/S1061920824040022Abstract 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 norming constants. We prove the uniqueness for solution of this problem for the first time. Moreover, we obtain an 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 the smoothness of the coefficients nor the self-adjointness of the operator. In addition, we establish the connection between McLaughlin's problem and Barcilon's three-spectra inverse problem. DOI 10.1134/S1061920824040022