Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Upper bounds for the ground state energy of the Laplacian with zero
  magnetic field on planar domains   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.

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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: mtrl-th