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Universal bounds and semiclassical estimates for eigenvalues of abstract Schrödinger operators
, 2008
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Trace identities for commutators, with applications to the distribution of eigenvalues
, 2009
"... 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 spe ..."
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Cited by 7 (1 self)
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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 LiebThirring inequalities. In the geometric context we derive a version of Reilly’s inequality bounding the eigenvalue λN+1 of the LaplaceBeltrami operator on an immersed manifold of dimension d by a universal constant times ‖h ‖ 2 ∞N 2/d.
On Riesz Means of Eigenvalues
, 2007
"... In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the RiemannLiouville fractional t ..."
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Cited by 6 (2 self)
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In this article we prove the equivalence of certain inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian with a classical inequality of Kac. Connections are made via integral transforms including those of Laplace, Legendre, Weyl, and Mellin, and the RiemannLiouville fractional transform. We also prove new universal eigenvalue inequalities and monotonicity principles for Dirichlet Laplacians as well as certain Schrödinger operators. At the heart of these inequalities are calculations of commutators of operators, sum rules, and monotonic properties of Riesz means. In the course of developing these inequalities we prove new bounds for the partition function and the spectral zeta function (cf. Corollaries 3.53.7) and conjecture about
A GEOMETRIC CHARACTERIZATION OF A SHARP HARDY INEQUALITY
, 1103
"... Abstract. In this paper, we prove that the distance function of an open connected set in R n+1 with a C 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C 2 ..."
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Cited by 4 (1 self)
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Abstract. In this paper, we prove that the distance function of an open connected set in R n+1 with a C 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex. We then prove that Hardy inequalities with a sharp constant hold on weakly mean convex C 2 domains. Moreover, we show that the weakly mean convexity condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of BrezisMarcus [7].
INEQUALITIES AND BOUNDS FOR THE EIGENVALUES OF THE SUBLAPLACIAN ON A STRICTLY PSEUDOCONVEX CR MANIFOLD
"... Abstract. We establish inequalities for the eigenvalues of the subLaplace operator associated with a pseudoHermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang [26] for the Dirichlet eigenvalues of the subLaplacian on a bounded domain ..."
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Cited by 2 (1 self)
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Abstract. We establish inequalities for the eigenvalues of the subLaplace operator associated with a pseudoHermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang [26] for the Dirichlet eigenvalues of the subLaplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known PaynePólyaWeinberger and Yang universal inequalities. hal00779283, version 1 28 Jan 2013 1.