Comptes Rendus
no. 4
Ordinary Differential Equations
When is a non-self-adjoint Hill operator a spectral operator of scalar type?

Presented by: 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.

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.

Received:
Accepted:
Published online:
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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