Solving An Inverse Problem For The Sturm-Liouville Operator With A Singular Potential By Yurko's Method

Abstract

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases.

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

“…Recall that, by using the spectra {λ s,jk } s≥1 , 1 ≤ k ≤ j ≤ n, one can uniquely construct the Weyl matrix M(λ) by formulas (35) and (32). Therefore, Theorem 2 implies the following corollary.…”

“…The constants C jk = 0 can be easily found by using the asymptotics of Lemma 2. Thus, given the eigenvalues {λ s,jk } s≥1 , one can find the characteristic functions ∆ jk (λ) and then construct all the nontrivial elements of the Weyl matrix by (32). In this way, Inverse Problem 4.2 is reduced to Inverse Problem 4.1.…”

“…Hryniv and Mykytyuk [20][21][22][23] extended the transformation operator method to the inverse problems for the Sturm-Liouville operators with singular potentials and so generalized a number of classical results to this class of operators. Afterwards, the theory of inverse spectral problems for the second-order differential operators with singular potentials was developed in the studies of Savchuk and Shkalikov [24,25], Freiling et al [26], Mykytyuk and Trush [27], Hryniv and Pronska [28,29], Eckhardt et al [30], Guliyev [31], Bondarenko [32][33][34], etc.…”

“…Classical results in this field were obtained for the Sturm-Liouville operators −y ′′ + q(x)y with integrable potentials q by using the famous transformation operator method (see the monographs [20][21][22][23] and references therein). For distributional potentials of classes W α 2 , α ≥ −1, inverse problems also have been studied fairly completely (see, e.g., [24][25][26][27][28][29][30][31]). However, inverse problems for higher-order (n > 2) differential operators are essentially different, because the transformation operator method is ineffective for them.…”

“…In recent years, the ideas of the method of spectral mappings were extended to operators generated by the differential expression (1.1) with τ n−1 = 0 (see [16-19, 26, 30]). In particular, in [26,30], the method of spectral mappings has been transferred to the Sturm-Liouville operators with distribution potentials of class W −1 2 [0, 1]. In [16], the uniqueness theorems of inverse spectral problems have been proved for the higher-order differential operators with distribution coefficients of the Mirzoev-Shkalikov class [1,2] on a finite interval.…”