Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Even order periodic operators on the real line   We consider $2p\ge 4$ order differential operator on the real line with a
periodic coefficients. The spectrum of this operator is absolutely continuous
and is a union of spectral bands separated by gaps. We define the Lyapunov
function, which is analytic on a p-sheeted Riemann surface. The Lyapunov
function has real or complex branch points. We prove the following results: (1)
The spectrum at high energy has multiplicity two. (2) Endpoints of all gaps are
periodic (or anti-periodic) eigenvalues or real branch points. (3) The spectrum
of operator has an infinite number of open gaps and there exists only a finite
number of non-real branch points for some specific coefficients (the generic
case). (4) The asymptotics of the periodic, anti-periodic spectrum and branch
points are determined at high energy.

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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con