Issue |
ESAIM: COCV
Volume 21, Number 3, July-September 2015
|
|
---|---|---|
Page(s) | 670 - 689 | |
DOI | https://doi.org/10.1051/cocv/2014043 | |
Published online | 13 May 2015 |
Magnetic spectral bounds on starlike plane domains
1
Department of Mathematics, University of Illinois,
Urbana, IL
61801,
USA
Laugesen@illinois.edu
2
Department of Mathematics, University of Oregon,
Eugene, OR
97403,
USA
Siudeja@uoregon.edu
Received:
5
January
2014
Revised:
3
June
2014
We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike
plane domains, under either Dirichlet or Neumann boundary conditions and assuming a
constant magnetic field in the transverse direction. Our main result says that
∑j=1n
Φ(λjA/G)
is maximal for a disk whenever Φ is concave increasing, n ≥ 1, the domain has area
A, and
λj is the
jth
Dirichlet eigenvalue of the magnetic Laplacian (i∇ +
(−x2,x1))2.
Here the flux β is constant, and the scale invariant factor
G penalizes
deviations from roundness, meaning G ≥ 1 for all domains and G = 1 for disks.
Mathematics Subject Classification: 35P15 / 35J20
Key words: Isoperimetric / spectral zeta / heat trace / partition function / Pauli operator
© EDP Sciences, SMAI, 2015
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