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

Abstract

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(λ_jA/G) is maximal for a disk whenever Φ is concave increasing, n ≥ 1, the domain has area A, and λ_j 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.