Magnetic spectral bounds on starlike plane domains
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