Inequalities for Means of Chords, with Application to Isoperimetric Problems |
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Authors: | Pavel Exner Evans M Harrell Michael Loss |
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Institution: | (1) Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, 25068 Řež near Prague, Czech Republic;(2) Doppler Institute, Czech Technical University, Břehová 7, 11519 Prague, Czech Republic;(3) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA |
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Abstract: | We consider a pair of isoperimetric problems arising in physics. The first concerns a Schrödinger operator in $L^2(\mathbb{R}^2)We consider a pair of isoperimetric problems arising in physics. The first concerns a Schr?dinger operator in
with an attractive interaction supported on a closed curve Γ, formally given by −Δ−αδ(x−Γ); we ask which curve of a given length maximizes the ground state energy. In the second problem we have a loop-shaped thread
Γ in
, homogeneously charged but not conducting, and we ask about the (renormalized) potential-energy minimizer. Both problems
reduce to purely geometric questions about inequalities for mean values of chords of Γ. We prove an isoperimetric theorem
for p-means of chords of curves when p ≤ 2, which implies in particular that the global extrema for the physical problems are always attained when Γ is a circle.
The letter concludes with a discussion of the p-means of chords when p > 2. |
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Keywords: | isoperimetric electrostatics Schr?dinger equation singular potential |
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