Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

Abstract

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

On recovering quadratic pencils with singular coefficients and entire functions in the boundary conditions

“…Consider the Weyl function M(𝜆) = − C(𝜋,𝜆) S(𝜋,𝜆) . In Bondarenko and Gaidel, 29 the following inverse problem was studied: Inverse Problem 2. Given {𝜃 k } k∈Z 0 and the values…”

“…Since {𝜃 k } k∈Z 0 are the zeros of S(𝜋, 𝜆), we uniquely construct the input data of inverse problem 2. In Bondarenko and Gaidel, 29 for inverse problem 2, a uniqueness theorem and an algorithm of solution were obtained in the singular case q ∈ W −1 2 (0, 𝜋) and p ∈ L 2 (0, 𝜋). In the regular case q ∈ L 2 (0, 𝜋) and p ∈ W 1 2 (0, 𝜋), the corresponding results were obtained earlier in previous studies.…”

“…6,14 Further, we reduce inverse problem 1 to inverse problem 2 and apply the results from Bondarenko and Gaidel. 29…”

“…Find p and q as the solution of inverse problem 2, see Algorithm 2.8 in Bondarenko and Gaidel. 29 6. Put 𝜔 = 1 𝜋 ∫ 𝜋 0 p(s) ds.…”

“…We note that inverse problems for pencil (1) with singular coefficients were studied in earlier studies. 12,15,29 In the present paper, we focus on the singular case and study a new type of inverse spectral problems for pencil (1), which deals with special boundary conditions involving arbitrary transcendental entire functions of the spectral parameter 𝑓 1 (𝜆) and 𝑓 2 (𝜆): 𝑦(0) = 0, 𝑓 1 (𝜆)𝑦 [1] (𝜋) + 𝑓 2 (𝜆)𝑦(𝜋) = 0.…”