Direct and inverse three‐point Sturm–Liouville problem with parameter‐dependent boundary conditions

Article type: Research Article

Authors: Pivovarchik, Vyacheslav

Affiliations: Department of Higher Mathematics, Odessa State Academy of Civil Engineering and Architecture, Didrihson str. 4, 65029 Odessa, Ukraine E‐mail: v.pivovarchik@paco.net

Abstract: The problem of small vibrations of a smooth inhomogeneous string damped at an interior point and fixed at the endpoints is reduced to a three‐point Sturm–Liouville boundary problem. This problem is considered as an eigenvalue problem for a nonmonic quadratic operator pencil of a special type with the spectrum located in the upper half‐plane of the spectral parameter. Concerning the corresponding inverse problem it is shown that the spectrum does not determine the potential of the Sturm–Liouville problem (and consequently the density of the string) uniquely. The conditions are given sufficient for a sequence of complex numbers to be the spectrum of the considered Sturm–Liouville problem with real‐valued potential which belongs to L2(0,a). In order to recover the potential uniquely the spectrum of the corresponding so to say truncated Sturm–Liouville problem is chosen as an additional information. Then the problem of recovering of the potential turns out to be overdetermined. The self‐consistency of the two spectra is discussed.

Journal: Asymptotic Analysis, vol. 26, no. 3-4, pp. 219-238, 2001