The magnetic D-to-N ground state energy obeys λ = α̂ b^{1/2} − α̂² + (1/3) max κ + o(1) in 2D, with a 3D limit given by the boundary infimum of λ^DN(ϑ(x)) |B(x)|^{1/2}.
Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --
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abstract
Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to the reader an overview of magnetic heat trace formulas through explicit examples. On the way we obtain new formulas and in particular we calculate explicitely some non local terms and logarithmic terms appearing in the Steklov heat trace asymptotics.
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Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains
The magnetic D-to-N ground state energy obeys λ = α̂ b^{1/2} − α̂² + (1/3) max κ + o(1) in 2D, with a 3D limit given by the boundary infimum of λ^DN(ϑ(x)) |B(x)|^{1/2}.