A semi-classical inverse problem II: reconstruction of the potential

Yves Colin de Verdière (1)

1 IF - Institut Fourier

This paper is the continuation of our work with Victor Guillemin; Victor and I proved that the Taylor expansion of the potential at a generic non degenerate critical point is determined by the semi-classical spectrum of the associated Schrödinger operator near the corresponding critical value. Here, I show that, under some genericity assumptions, the potential of the 1D Schroedinger operator is determined by its semi-classical spectrum. Moreover, there is an explicit reconstruction. This paper is strongly related to a paper of David Gurarie (J. Math. Phys. 36:1934--1944 (1995)).

Mots clés: inverse spectral problem; trace formulae; Schrödinger operator; semi-classics

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Théorie spectrale [math.SP]

Yves Colin de Verdière. A semi-classical inverse problem II: reconstruction of the potential. 2008. ⟨hal-00251590⟩