Commutators, Eigenvalue Gaps, and Mean Curvature in the Theory of Schrödinger Operators

DOI10.1080/03605300500532889zbMath1387.35136arXivmath/0312372OpenAlexW2047771510MaRDI QIDQ3436042

Evans M. II. Harrell

Publication date: 8 May 2007

Published in: Communications in Partial Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0312372

zbMATH Keywords

eigenvaluemean curvatureLaplace-BeltramiSchrödinger operations

Mathematics Subject Classification ID

General theory of partial differential operators (47F05) Elliptic equations on manifolds, general theory (58J05) Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Schrödinger operator, Schrödinger equation (35J10) Applications of differential geometry to physics (53Z05)

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A Jacobi spectral method for computing eigenvalue gaps and their distribution statistics of the fractional Schrödinger operator ⋮ Estimates for eigenvalues of the elliptic operator in divergence form on Riemannian manifolds ⋮ Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold ⋮ Estimates for eigenvalues of fourth-order elliptic operators in divergence form ⋮ Payne-Polya-Weinberger, Hile-Protter and Yang's inequalities for Dirichlet Laplace eigenvalues on integer lattices ⋮ A geometric characterization of a sharp Hardy inequality ⋮ On eigenvalues of a system of elliptic equations and of the biharmonic operator ⋮ Inequalities for eigenvalues of the drifting Laplacian on Riemannian manifolds ⋮ Inequalities for eigenvalues of a clamped plate problem ⋮ Inequalities for eigenvalues of the biharmonic operator with weight on Riemannian manifolds ⋮ Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds ⋮ Inequalities for eigenvalues of elliptic operators in divergence form on Riemannian manifolds ⋮ Universal inequalities for the eigenvalues of Laplace and Schrödinger operators on submanifolds ⋮ Extrinsic eigenvalue estimates of Dirac operators on Riemannian manifolds ⋮ On Riesz Means of Eigenvalues ⋮ Eigenvalue inequalities for Klein-Gordon operators ⋮ Estimates for eigenvalues on Riemannian manifolds ⋮ Trace identities for commutators, with applications to the distribution of eigenvalues ⋮ Universal bounds for eigenvalues of a buckling problem II ⋮ Estimates for eigenvalues of the poly-Laplacian with any order in a unit sphere

Cites Work

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• On the Laplace operator penalized by mean curvature
• On the second eigenvalue of the Laplace operator penalized by curvature
• The universal eigenvalue bounds of Payne-Pólya-Weinberger, Hile-Protter, and H. C. Yang
• A unified approach to universal inequalities for eigenvalues of elliptic operators
• Commutators, spectral trace identities, and universal estimates for eigenvalues
• Some geometric bounds on eigenvalue gaps
• On trace identities and universal eigenvalue estimates for some partial differential operators
• CURVATURE-INDUCED BOUND STATES IN QUANTUM WAVEGUIDES IN TWO AND THREE DIMENSIONS

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