Inverse spectral results for Schrödinger operators on the unit interval with partial informations given on the potentials

We pursue the analysis of the Schrödinger operator on the unit interval in inverse spectral theory initiated in [AR]. Whereas the potentials in [AR] belong to L with their difference in L (1 ≤ p < ∞) we consider here potentials in W k,1 spaces having their difference in W k,p where 1 ≤ p ≤ +∞, k ∈ {0, 1, 2}. It is proved that two potentials in W ([0, 1]) being equal on [a, 1] are also equal on [0, 1] if their difference belongs to W ([0, a]) and if the number of their common eigenvalues is sufficiently high. Naturally, this number decreases as the parameter a decreases and as the parameters k and p are increasing. ∗Laboratoire de Mathématiques EDPPM, FRE-CNRS 3111, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France. [email protected] †Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351 cours de la libération, 33405 Talence Cedex, France. [email protected] ‡Laboratoire de Mathématiques EDPPM, FRE-CNRS 3111, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France. [email protected] 1 ha l-0 03 85 83 8, v er si on 1 20 M ay 2 00 9 Author manuscript, published in "Journal of Mathematical Physics 50 (2009) 033505" DOI : 10.1063/1.3087426