An application of approximation theory by nonlinear manifolds in Sturm Liouville inverse problems

We give here some negative results in Sturm-Liouville inverse theory. Namely, we prove that one cannot approach any of the potentials with m + 1 integrable derivatives on {\bb R}^+ by a fixed ω-parametric analytic family better than of order (ωln ω)^-(m+1). Next, we prove an estimation of the eigenvalues and characteristic constants of a Sturm-Liouville operator and some properties of the solution of a certain integral equation of Gelfand-Levitan type. This information allows us to deduce from Henkin and Novikova (1996 Stud. Appl. Math. 97 17-52) an ω-parametric analytic approximation formula giving the reconstruction of any negative potential with m + 1 integrable derivatives with precision of order ω^-m, in terms of eigenvalues and characteristic constants of its associated Sturm-Liouville operator.