Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectrum decomposition of translation operators in periodic waveguide   Scattering problems in periodic waveguides are interesting but also
challenging topics in mathematics, both theoretically and numerically. Due to
the existence of eigenvalues, the unique solvability of these problems is not
always guaranteed. To obtain a unique solution that is "physically meaningful",
the limiting absorption principle (LAP) is a commonly used method. LAP assumes
that the limit of a family of solutions with absorbing media converges, as the
absorption parameter tends to 0, and the limit is the "physically meaningful
solution". It is also called the LAP solution in this paper. It has been proved
that the LAP holds for periodic waveguides in [Hoa11]. In this paper, we
consider the spectrum decomposition of periodic translation operators. With the
curve integral formulation and a generalized Residue theorem, the operator is
explicitly described by its eigenvalues and generalized eigenfunctions, which
are closely related to Bloch wave solutions. Then the LAP solution is
decomposed into generalized eigenfunctions. This gives a better understanding
of structure of the scattered fields.

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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: alg-geom