TY - JOUR
T1 - Sums of Magnetic Eigenvalues are Maximal on Rotationally Symmetric Domains
AU - Laugesen, Richard S.
AU - Liang, Jian
AU - Roy, Arindam
N1 - Funding Information:
This work was partially supported by a grant from the Simons Foundation (#204296 to Richard Laugesen). The authors also acknowledge support from the National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students”. We thank Bart lomiej Siudeja for creating Fig. 1, and Rupert Frank for alleviating our ignorance on certain issues.
PY - 2012/5
Y1 - 2012/5
N2 - 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).
AB - 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).
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U2 - 10.1007/s00023-011-0142-z
DO - 10.1007/s00023-011-0142-z
M3 - Article
AN - SCOPUS:84859868871
SN - 1424-0637
VL - 13
SP - 731
EP - 750
JO - Annales Henri Poincare
JF - Annales Henri Poincare
IS - 4
ER -