Linear differential operators with distribution coefficients of various singularity orders

Abstract

In this paper, the linear differential expression of order n ≥ 2 with distribution coefficients of various singularity orders is considered. We obtain the associated matrix for the regularization of this expression. Furthermore, we present new statements of inverse spectral problems that consist in the recovery of differential operators from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these two inverse problems are proved by developing the method of spectral mappings. To the best of the author's knowledge, inverse problems for higher order differential operators with distribution coefficients on the half-line have not been studied before. For the finite interval case, we for the first time consider the issue of recovering coefficients of the boundary conditions.

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

“…2.6, which is more convenient for studying the asymptotics of solutions. This section is based on the results of [2,4,14,16]. The associated matrix F(x) is defined by the coefficients ( ) n−2 =0 of the differential expression Eq.…”

“…However, for differential operators with distribution coefficients, it is more convenient to consider the divergent form Eq. 1.1 following [2][3][4]. For regular differential operators Eq.…”

“…In particular, the approach of [14] has been applied to the differential expression of form Eq. 1.1 in [4]. 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.…”

“…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.…”

Inverse spectral problem for the third-order differential equation

“…Recently, the study of inverse spectral problems began for higher-order differential operators with distribution coefficients. In [24,25], the uniqueness theorems for the recovery of such operators on a finite interval and on the half-line have been proved. In [26], a constructive approach for solving inverse spectral problems has been developed.…”