Reconstruction of Higher-Order Differential Operators by Their Spectral Data

Abstract

This paper is concerned with inverse spectral problems for higher-order (n>2) ordinary differential operators. We develop an approach to the reconstruction from the spectral data for a wide range of differential operators with either regular or distribution coefficients. Our approach is based on the reduction of an inverse problem to a linear equation in the Banach space of bounded infinite sequences. This equation is derived in a general form that can be applied to various classes of differential operators. The unique solvability of the linear main equation is also proved. By using the solution of the main equation, we derive reconstruction formulas for the differential expression coefficients in the form of series and prove the convergence of these series for several classes of operators. The results of this paper can be used for the constructive solution of inverse spectral problems and for the investigation of their solvability and stability.

Spectral Data Asymptotics for the Higher-Order Differential Operators With Distribution Coefficients

“…In addition, we obtain the analogs of Theorems 1.1 and 1.2 for the weight numbers defined in the "Asymptotics of weight numbers" section. The weight numbers together with the eigenvalues are used as spectral data for recovering higher-order differential operators with distribution coefficients in [1]. The paper is organized as follows.…”

“…The paper aims to study spectral data asymptotics for the differential equation subject to the separated boundary conditions. The results of this paper are applied in [1] to the inverse problem theory and also have a separate significance.…”

“…It is worth mentioning that, in [2,3,14], the coefficients at y (n) and y (n−1) in the differential expression can be arbitrary functions of some classes. In this paper, we confine ourselves to the coefficients 1 and 0 at y (n) and y (n−1) , respectively, because this case is natural for the inverse problem theory (see [1,4,15]). Let us briefly describe the regularization of the differential expression Eq.…”

“…By using the quasi-derivatives Eq. 1.3 is reduced to the equivalent system where y(x) = col(y [0] (x), y [1] (x), … , y [n−1] (x)) , Λ is the (n × n) matrix whose entry at the position (n, 1) equals and all the other entries equal 0. Define the boundary conditions…”

“…The first result of this paper is the following theorem, which describes the asymptotic behaviour of the eigenvalues. In the inverse problem theory [1], it is convenient to recover the coefficients n−2 , n−3 , ..., 1 , 0 one-by-one. Therefore, the question arises:…”