Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: First-order nonlinear eigenvalue problems involving functions of a
  general oscillatory behavior   Eigenvalue problems arise in many areas of physics, from solving a classical
electromagnetic problem to calculating the quantum bound states of the hydrogen
atom. In textbooks, eigenvalue problems are defined for linear problems,
particularly linear differential equations such as time-independent
Schr\"odinger equations. Eigenfunctions of such problems exhibit several
standard features independent of the form of the underlying equations. As
discussed in Bender \emph{et al}
[\href{http://dx.doi.org/10.1088/1751-8113/47/23/235204}{J.~Phys.~A 47, 235204
(2014)}], separatrices of nonlinear differential equations share some of these
features. In this sense, they can be considered eigenfunctions of nonlinear
differential equations, and the quantized initial conditions that give rise to
the separatrices can be interpreted as eigenvalues. We introduce a first-order
nonlinear eigenvalue problem involving a general class of functions and obtain
the large-eigenvalue limit by reducing it to a random walk problem on a
half-line. The introduced general class of functions covers many special
functions such as the Bessel and Airy functions, which are themselves solutions
of second-order differential equations. For instance, in a special case
involving the Bessel functions of the first kind, i.e., for $y'(x)=J_\nu(xy)$,
we show that the eigenvalues asymptotically grow as $2^{41/42} n^{1/4}$. We
also introduce and discuss nonlinear eigenvalue problems involving the
reciprocal gamma and the Riemann zeta functions, which are not solutions to
simple differential equations. With the reciprocal gamma function, i.e., for
$y'(x)=1/\Gamma(-xy)$, we show that the $n$th eigenvalue grows factorially fast
as $\sqrt{(1-2n)/\Gamma(r_{2n-1})}$, where $r_k$ is the $k$th root of the
digamma function.

11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
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The categories of this paper are: astro-ph
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
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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: solv-int
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The categories of this paper are: stat
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: econ