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Harrell, Evans M.

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Commutator Bounds for Eigenvalues of Some Differential Operators

1994-03 , Harrell, Evans M. , Michel, Patricia L.

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Commutator Bounds for Eigenvalues, with Applications to Spectral Geometry

1994-03 , Harrell, Evans M. , Michel, Patricia L.

We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace-Beltrami operators on manifolds. We significantly improve earlier bounds of Yang and Yau, Li, and Harrell.

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On the Second Eigenvalue of the Laplace Operator Penalized by Curvature

1994 , Harrell, Evans M.

Consider the operator - ∇^2 - q(κ), where - ∇^2 is the (positive) Laplace-Beltrami operator on a closed manifold of the topological type of the two-sphere S^2 and q is a symmetric non-negative quadratic form in the principal curvatures. Generalizing a well-known theorem of J. Hersch for the Laplace-Beltrami operator alone, it is shown in this note that the second eigenvalue λ [1] is uniquely maximized, among manifolds of fixed area, by the true sphere.

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