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Harrell, Evans M.

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Pointwise Control of Eigenfunctions on Quantum Graphs

2016-10-09 , Harrell, Evans M.

Pointwise bounds on eigenfunctions are useful for establishing localization of quantum states, and they have implications for the distribution of eigenvalues and for physical properties such as conductivity. In the low-energy regime, localization is associated with exponential decrease through potential barriers. We adapt the Agmon method to control this tunneling effect for quantum graphs with Sobolev and pointwise estimates. It turns out that as a generic matter, the rate of decay is controlled by an Agmon metric related to the classical Liouville- Geen approximation for the line, but more rapid decay is typical, arising from the geometry of the graph. In the high-energy regime one expects states to oscillate but to be dominated by a 'landscape function' in terms of the potential and features of the graph. We discuss the construction of useful landscape functions for quantum graphs.

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Expanding the Reach of Your Research: An Open Forum on Authorship and Your Intellectual Property

2010-10-21 , Harrell, Evans M. , Beck, Joseph M. , Herrington, TyAnna K. , Harvey, Stephen C. , Bobick, Aaron

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Commutator Bounds for Eigenvalues of Some Differential Operators

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

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Spectral and variational problems of mathematical physics

1990 , Harrell, Evans M.

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On a transformation of Bohl and its discrete analogue

2013 , Harrell, Evans M. , Wong, Manwah Lilian

Fritz Gesztesy’s varied and prolific career has produced many transformational contributions to the spectral theory of one-dimensional Schrödinger equations. He has often done this by revisiting the insights of great mathematical analysts of the past, connecting them in new ways, and reinventing them in a thoroughly modern context. In this short note we recall and relate some classic transformations that figure among Fritz Gestesy’s favorite tools of spectral theory, and indeed thereby make connections among some of his favorite scholars of the past, Bohl, Darboux, and Green. After doing this in the context of one-dimensional Schrödinger equations on the line, we obtain some novel analogues for discrete one-dimensional Schrödinger equations.

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Spectra, Geometry and Asymptotics of Some Differential Equations of Mathematical Physics

2002-06-01 , Harrell, Evans M.

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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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Promotion and Tenure Panel Discussion

2011-04-15 , Harrell, Evans M. , Sokolik, Irina N. , Friedman, Kathryn Muir , Wiesenfeld, Kurt

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Spectral and asymptotic problems of mathematical physics

1999 , Harrell, Evans M.

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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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