Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Zero modes of quantum graph Laplacians and an index theorem   We study zero modes of Laplacians on compact and non-compact metric graphs
with general self-adjoint vertex conditions. In the first part of the paper the
number of zero modes is expressed in terms of the trace of a unitary matrix
$\mathfrak{S}$ that encodes the vertex conditions imposed on functions in the
domain of the Laplacian. In the second part a Dirac operator is defined whose
square is related to the Laplacian. In order to accommodate Laplacians with
negative eigenvalues it is necessary to define the Dirac operator on a suitable
Kre\u{\i}n space. We demonstrate that an arbitrary, self-adjoint quantum graph
Laplacian admits a factorisation into momentum-like operators in a
Kre\u{\i}n-space setting. As a consequence, we establish an index theorem for
the associated Dirac operator and prove that the zero-mode contribution in the
trace formula for the Laplacian can be expressed in terms of the index of the
Dirac operator.

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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: stat