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arXiv · 1401.0850

Magnetic spectral bounds on starlike plane domains

R. S. Laugesen·B. A. Siudeja
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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 $\sum_{j=1}^n \Phi \big( \lambda_j A/G \big)$ is maximal for a disk whenever $\Phi$ is concave increasing, $n \geq 1$, the domain has area $A$, and $\lambda_j$ is the $j$-th Dirichlet eigenvalue of the magnetic Laplacian $\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2$. Here the flux $\beta$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G \geq 1$ for all domains and $G=1$ for disks.

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R. S. Laugesen, B. A. Siudeja. 2014-01-04. Magnetic spectral bounds on starlike plane domains. https://arxiv.org/abs/1401.0850

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