Comptes Rendus
no. 4
Ordinary Differential Equations
When is a non-self-adjoint Hill operator a spectral operator of scalar type?
[Quand un opérateur de Hill non-autoadjoint est-il un operateur spectral de type scalaire ?]

Présenté par : Peter Lax

Fritz Gesztesy1 ; Vadim Tkachenko2
1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
2 Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel
Comptes Rendus. Mathématique, Volume 343 (2006) no. 4, pp. 239-242.

Nous dérivons des conditions nécessaires et suffisantes pour qur l'opérateur de Schrödinger (i.e., l'opérateur de Hill) H=−d2/dx2+V dans L2(R) soit un opérateur spectral de type scalaire. Les conditions montrent que cette propriétés ne dépend pas des propriétés différentielles (ou analytiques) du potentiel V.

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H=−d2/dx2+V in L2(R) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.06.014
Fritz Gesztesy 1 ; Vadim Tkachenko 2

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
2 Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105, Israel 
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Fritz Gesztesy; Vadim Tkachenko. When is a non-self-adjoint Hill operator a spectral operator of scalar type?. Comptes Rendus. Mathématique, Volume 343 (2006) no. 4, pp. 239-242. doi : 10.1016/j.crma.2006.06.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.06.014/

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