Inverse spectral problems for coupled oscillating systems

Abstract

We study an inverse spectral problem for a compound oscillating system consisting of a singular string and N masses joined by springs. The operator A corresponding to this system acts in L 2 (0, 1) × C N and is composed of a Sturm-Liouville operator in L 2 (0, 1) with a distributional potential and a Jacobi matrix in C N , which are coupled in a special way. We solve the inverse spectral problem for the operator A and describe explicitly the set of spectral data. We also exhibit a connection to related Sturm-Liouville problems with boundary conditions depending rationally on the spectral parameter.

Schrödinger operators with distributional potentials and boundary conditions dependent on the eigenvalue parameter

“…Let ν 0 be the smallest zero of ϕ(π, λ). Dividing both sides of the identity ϕ(π, λ)ϕ [1] s (π, µ) − ϕ [1] s (π, λ)ϕ(π, µ)…”

“…At the end of the last millennium, Savchuk and Shkalikov [26] initiated the study of boundary value problems associated with differential equations of the form − y [1] s ′ (x) − s(x)y [1] s (x) − s 2 (x)y(x) = λy(x) (1.1)…”

“…where s ∈ L 2 (0, π) and y [1] s (x) := y ′ (x) − s(x)y(x) denotes the quasi-derivative of y with respect to s (the subscript is usually omitted from the notation, but we keep it because in this paper we will consider several potentials simultaneously). This equation formally corresponds to the one-dimensional Schrödinger equation with the distributional potential s ′ ∈ W −1 2 (0, π).…”

“…In this paper we study various direct and inverse spectral problems for boundary value problems generated by the equation (1.1) and the boundary conditions y [1] s (0) y(0) = −f (λ), y [1] s (π) y(π) = F (λ), (1.2) where…”

An inverse three spectra problem for Sturm–Liouville operators

“…In [4], the authors further generalized the uniqueness theorem by σ (0, 1; h 0 , h 1 ; q), σ (0, c; h 0 , h c ; q) and σ (c, 1; h c , h 1 ; q) with c ∈ (0, 1) and h c ∈ R ∪ {∞}. In the past years, the inverse three spectra problem has been investigated by several authors (see [5][6][7][8][9][10][11][12][13][14][15] and the references therein). The known results contain the existence, the uniqueness, the numerical scheme, the reconstructing formula, and so on.…”