For a magnetic field vanishing to finite order, the ground state energies scale like β^(2/(κ*+2)) for Dirichlet/Neumann and β^(1/(κ0+2)) for Dirichlet-to-Neumann, with one-term asymptotics under non-degeneracy conditions.
Attar, The ground state energy of the two dimensional Ginzburg-Landa u functional with variable magnetic field , Ann
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The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain
For a magnetic field vanishing to finite order, the ground state energies scale like β^(2/(κ*+2)) for Dirichlet/Neumann and β^(1/(κ0+2)) for Dirichlet-to-Neumann, with one-term asymptotics under non-degeneracy conditions.