IOP Publishing, Inverse Problems, 1(23), p. 99-121, 2006
DOI: 10.1088/0266-5611/23/1/005
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Given a measure $m$ on the real line or a finite interval, the "cubic string" is the third order ODE $-ϕ'''=zmϕ$ where $z$ is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a nonselfadjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis--Procesi equation. We solve the spectral and inverse spectral problem for the case of $m$ being a finite positive discrete measure. In particular, explicit determinantal formulas for the measure $m$ are given. These formulas generalize Stieltjes' formulas used by Krein in his study of the corresponding second order ODE $-ϕ''=zmϕ$.