Geodesic
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

Gesztesy, Fritz1 ; Tkachenko, Vadim2
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, Tome 343 (2006) no. 4, pp. 239-242.

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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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.06.014

Gesztesy, Fritz 1 ; Tkachenko, Vadim 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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Gesztesy, Fritz; Tkachenko, Vadim. When is a non-self-adjoint Hill operator a spectral operator of scalar type?. Comptes Rendus. Mathématique, Tome 343 (2006) no. 4, pp. 239-242. doi : 10.1016/j.crma.2006.06.014. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2006.06.014/

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