Inverse Spectral Problem for the Third-Order Differential Equation

Reconstruction of Higher-Order Differential Operators by Their Spectral Data

“…Theorem 1.3 generalizes the previous results of [12] for n = 2 and of [35] for n = 3. However, for n ≥ 4, to the best of the author's knowledge, Theorem 1.3 is the first existence result for the inverse problem solution in the case of distribution coefficients.…”

“…In particular, for multiple eigenvalues in the case n = 2 the generalized weight numbers were defined in [50,51]. For n = 3 and λ l,1 = λ l,2 , the additional weight numbers γ l were used in [35]. For higher orders n, the situation becomes much more complicated, so in this paper we confine ourselves to the class W .…”

“…This has motivated a number of researchers to study solutions and spectral theory for such equations (see the bibliography in the recent papers [30,31]). Inverse spectral problems for higher-order differential operators with distribution coefficients have been investigated in [31][32][33][34][35]. In particular, the uniqueness theorems have been proved in [31][32][33].…”

“…The paper [34] is concerned with a reconstruction approach, based on developing the ideas of the method of spectral mappings. In [35], the necessary and sufficient conditions for the inverse problem solvability have been obtained for a third-order differential equation.…”

“…Furthermore, local solvability and stability in the L 2 -norm were formulated without proofs as Theorems 2.3.4 and 2.5.3 in [22]. For distribution coefficients, local solvability and stability theorems have been proved in [12] for n = 2 and in [35] for n = 3. However, to the best of the author's knowledge, there were no results in this direction for n ≥ 4.…”