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 the new statements of inverse spectral problems that consist in the recovery of differential operators with distribution coefficients from the Weyl matrix on the half-line and on a finite interval. The uniqueness theorems for these inverse problems are proved by developing the method of spectral mappings.

Inverse Spectral Problems for Arbitrary-Order Differential Operators with Distribution Coefficients

“…q k (x)y (k) , (1.4) where (q k ) n−2 k=0 are some integrable functions. However, for differential operators with distribution coefficients, it is more convenient to consider the divergent form (1.1) following [2][3][4].…”

“…Vladimirov [14] has obtained an alternative construction, which can be used a wider class of differential operators than the results of [2,3]. In particular, the approach of [14] has been applied to the differential expression of form (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]).…”

“…In this section, equation (1.3) is reduced to the system (1.6) and then to the form (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) defined by the coefficients (σ ν ) n−2 ν=0 of the differential expression (1.1) as follows.…”

“…then ℓ n (y) is a regular function and ℓ n (y) = y [n] (see [4,Theorem 2.2]). Then, instead of equation ( 1.3), one can consider the system (1.6).…”