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

Laurent Amour (1) , Jérémy Faupin (2) , Thierry Raoux (1)

1 LMR - Laboratoire de Mathématiques de Reims
2 IMB - Institut de Mathématiques de Bordeaux

We pursue the analysis of the Schrödinger operator on the unit interval in inverse spectral theory initiated in the work of Amour and Raoux ["Inverse spectral results for Schrödinger operators on the unit interval with potentials in $L^p$ spaces", Inverse Probl. 23, 2367 (2007)]. While the potentials in the work of Amour and Raoux belong to $L^1$ with their difference in $L^p$, $1 \le p < +\infty$, we consider here potentials in $W^{k,1}$ spaces having their difference in $W^{k, p}$, where $1 \le p \le + \infty$, $k \in \{0 , 1 , 2\}$. It is proved that two potentials in $W^{k,1}([0,1])$ being equal on $[a,1]$ are also equal on $[0,1]$ if their difference belongs to $W^{k, p}([0,a])$ 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 $p$ increase.

Domaines: Physique mathématique [math-ph]

Laurent Amour, Jérémy Faupin, Thierry Raoux. Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials. Journal of Mathematical Physics, 2009, 50, pp.033505. ⟨10.1063/1.3087426⟩. ⟨hal-00385838⟩