Sums of Magnetic Eigenvalues are Maximal on Rotationally Symmetric Domains

Richard S. Laugesen, Jian Liang, Arindam Roy

Research output: Contribution to journalArticle


The sum of the first n ≥ 1 energy levels of the planar Laplacian with constant magnetic field of given total flux is shown to be maximal among triangles for the equilateral triangle, under normalization of the ratio (moment of inertia)/(area) 3 on the domain. The result holds for both Dirichlet and Neumann boundary conditions, with an analogue for Robin (or de Gennes) boundary conditions too. The square similarly maximizes the eigenvalue sum among parallelograms, and the disk maximizes among ellipses. More generally, a domain with rotational symmetry will maximize the magnetic eigenvalue sum among all linear images of that domain. These results are new even for the ground state energy (n = 1).

Original languageEnglish (US)
Pages (from-to)731-750
Number of pages20
JournalAnnales Henri Poincare
Issue number4
StatePublished - May 1 2012

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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