Local solvability and stability of inverse problems for Sturm-Liouville operators with a discontinuity

Abstract

Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm for solving the inverse problems. The key role in our method is played by the Riesz-basis property of a special vectorfunctional system in a Hilbert space. In addition, we obtain a new uniqueness theorem for recovering the potential on a part of the interval, by using a fractional part of the spectrum.

An inverse problem for the Sturm-Liouville pencil with arbitrary entire functions in the boundary condition

“…Proof. One can prove this lemma similarly to [6,Lemma 3] and [42,Lemma 5], relying on the relations (3.11), (3.12), (4.10), (4.11). The only difference is, that the functions K(t) and N (t) are complex-valued, so the functions η j (λ), j = 0, 1, may have multiple zeros.…”

“…• partial inverse problems for Sturm-Liouville operators with discontinuities [23,38,41,42]; • partial inverse problems for quantum graphs [5,7,8,40,43].…”

“…In this paper, we prove the uniqueness theorem, local solvability and stability for Inverse Problem 1.1, and also provide a constructive algorithm for solution of this problem. Our technique develops the ideas of [5,6,42]. In order to solve the inverse problem, we reduce it to the classical Sturm-Liouville problem with boundary conditions independent of λ.…”

“…Suppose that the potential q(x) is known on (a, 1), a ≤ d, and our goal is to recover the potential on the remaining part of the interval by a part of the spectrum. Such partial inverse problems have been studied in [23,38,41,42].…”

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

“…Then, the eigenvalues of Equations coincide with the eigenvalues of Equations , where f 1 ( λ ):= ψ ( π , λ ), f 2 ( λ )=− ψ ′ ( π , λ ). Note that these functions f j ( λ ), j =1,2, can be determined by the given potential q ( x ), x ∈( π ,2 π ).In recent years, Inverse Problem was generalized to the case when the potential is known not on the half but on an arbitrary part ( a ,2 π ) of the interval (see Gesztesy and Simon and Horváth), to discontinuous Sturm–Liouville operators (see Hald, Shieh et al, and Yang and Bondarenko) and to differential operators on metric graphs (see other studies). Differential operators with discontinuities arise in geophysical models of the Earth oscillations (see Anderssen and Lapwood and Usami) and in electronics for constructing parameters of heterogeneous electronic lines with desirable technical characteristics (see Meschanov and Feldstein).…”