Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A criterion for the existence of zero modes for the Pauli operator with
  fastly decaying fields   We consider the Pauli operator in $\mathbb R^3$ for magnetic fields in
$L^{3/2}$ that decay at infinity as $|x|^{-2-\beta}$ with $\beta > 0$. In this
case we are able to prove that the existence of a zero mode for this operator
is equivalent to a quantity $\delta(\mathbf B)$, defined below, being equal to
zero. Complementing a result from [Balinsky, Evans, Lewis (2001)], this implies
that for the class of magnetic fields considered, Sobolev, Hardy and CLR
inequalities hold whenever the magnetic field has no zero mode.

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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
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The categories of this paper are: stat
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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: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: econ
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics