Abstract. We introduce a variational algorithm, which solves the classical inverse Sturm-Liouville problem when two spectra are given. In contrast to other approaches, it recovers the potential as well as the boundary conditions without a priori knowledge of the mean of the potential. Numerical examples show that the algorithm works quite reliable, even in the presence of noise. A proof of the ...

Abstract. We present a variational algorithm for solving the classical inverse Sturm-Liouville problem in one dimension when two spectra are given. All critical points of the least squares functional are at global minima, which justifies minimization by a (conjugate) gradient descent algorithm. Numerical examples show that the resulting algorithm works quite reliable without tuning for particul...

For any positive integer n and any given n distinct real numbers we construct a Sturm-Liouville problem whose spectrum is precisely the given set of n numbers. Such problems are of Atkinson type in the sense that the weight function or the reciprocal of the leading coefficient is identically zero on at least one subinterval.

We consider the nonlinear Sturm-Liouville problem −u′′(t) + f(u(t), u′(t)) = λu(t), u(t) > 0, t ∈ I := (−1/2, 1/2), u(±1/2) = 0, where f(x, y) = |x|p−1x − |y|m, p > 1, 1 ≤ m < 2 are constants and λ > 0 is an eigenvalue parameter. To understand well the global structure of the bifurcation branch of positive solutions in R+ ×Lq(I) (1 ≤ q < ∞) from a viewpoint of inverse problems, we establish the...

We discuss inverse spectral theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type τf = 1 r ( − ( p[f ′ + sf ] )′ + sp[f ′ + sf ] + qf ) , where the coefficients p, q, r, s are Lebesgue measurable on (a, b) with p−1, q, r, s ∈ Lloc((a, b); dx) and real-valued with p 6= 0 and r > 0 almost everywhere on (a, ...

In this paper, we consider the negative order Korteweg–de Vries equation with a self-consistent integral source. It is shown that negative-order source can be integrated by method of inverse spectral problem. The evolution data Sturm–Liouville operator periodic potential associated solution determined. obtained results make it possible to apply problem solve in class functions.

For the Sturm-Liouville operator L =L,: .VP+ -p” +pv one seeks to reconstruct the coefficient p from knowledge of the sequence of eigen-frequencies (Aj with LJ? =Ajn for some y j # 0). An implementable scheme is: for some N determine pnr so (approximately) pN has minimum norm with eigen-frequencies { A , . . . . .IN/ as given. This is the method ot’ ’generalised interpolation‘ and is shown to g...