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- W3041078307 abstract "We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential $1$-form (hence, with zero magnetic field) acting on complex functions of a planar domain $Omega$, with magnetic Neumann boundary conditions. It is well-known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $epsilon$-net. In the last part we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one." @default.
- W3041078307 created "2020-07-16" @default.
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- W3041078307 date "2020-07-09" @default.
- W3041078307 modified "2023-10-18" @default.
- W3041078307 title "Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains" @default.
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