Sums of Laplace eigenvalues: Rotations and tight frames in higher dimensions

R. S. Laugesen, B. A. Siudeja

Research output: Contribution to journalArticle

Abstract

The sum of the first n ≥ 1 eigenvalues of the Laplacian is shown to be maximal among simplexes for the regular simplex (the regular tetrahedron, in three dimensions), maximal among parallelepipeds for the hypercube, and maximal among ellipsoids for the ball, provided the volume and moment of inertia of an auxiliary body are suitably normalized. This result holds for Dirichlet, Robin, and Neumann eigenvalues. Additionally, the cubical torus is shown to be maximal among flat tori. The proof involves euclidean tight frames generated by orbits of the rotation group of the extremal domain. Among general convex domains, the ball is conjectured to maximize sums of Neumann eigenvalues, provided the volume and moment of inertia of the polar dual are suitably normalized.

Original languageEnglish (US)
Article number093703
JournalJournal of Mathematical Physics
Volume52
Issue number9
DOIs
StatePublished - Sep 23 2011

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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