TRACE IDENTITIES FOR COMMUTATORS, WITH APPLICATIONS TO THE DISTRIBUTION OF EIGENVALUES
Identifieur interne : 000D59 ( Main/Curation ); précédent : 000D58; suivant : 000D60TRACE IDENTITIES FOR COMMUTATORS, WITH APPLICATIONS TO THE DISTRIBUTION OF EIGENVALUES
Auteurs : Evans M. Ii Harrell [États-Unis] ; Joachim Stubbe [Suisse]Source :
- Transactions of the American Mathematical Society [ 0002-9947 ] ; 2011.
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- Pascal (Inist)
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Abstract
We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue λN+1 in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrödinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue λN+1 of the Laplace-Beltrami operator by a universal constant times ⁄h⁄2∞N2/d.
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<profileDesc><textClass><keywords scheme="KwdEn" xml:lang="en"><term>Distribution</term>
<term>Distribution function</term>
<term>Eigenvalue</term>
<term>Inequality</term>
<term>Lower bound</term>
<term>Manifold</term>
<term>Monotonicity</term>
<term>Schrödinger operator</term>
<term>Statistical moment</term>
<term>Sum rule</term>
<term>Trace</term>
</keywords>
<keywords scheme="Pascal" xml:lang="fr"><term>Trace</term>
<term>Distribution</term>
<term>Fonction répartition</term>
<term>Valeur propre</term>
<term>Règle somme</term>
<term>Opérateur Schrödinger</term>
<term>Variété mathématique</term>
<term>Borne inférieure</term>
<term>Monotonie</term>
<term>Moment statistique</term>
<term>60E05</term>
<term>65F15</term>
<term>65H17</term>
<term>Opérateur périodique</term>
<term>35J10</term>
<term>47A10</term>
<term>62E17</term>
<term>Opérateur Laplace Beltrami</term>
<term>Inégalité</term>
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<front><div type="abstract" xml:lang="en">We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schrödinger operators and Schrödinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue λ<sub>N+1</sub>
in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the case of a Schrödinger operator on an immersed manifold of dimension d, we derive a version of Reilly's inequality bounding the eigenvalue λ<sub>N+1</sub>
of the Laplace-Beltrami operator by a universal constant times ⁄h⁄<sup>2</sup>
∞N<sup>2/d</sup>
.</div>
</front>
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