Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotics with respect to the spectral parameter and Neumann series of
  Bessel functions for solutions of the one-dimensional Schr\"odinger equation   A representation for a solution $u(\omega,x)$ of the equation
$-u"+q(x)u=\omega^2 u$, satisfying the initial conditions $u(\omega,0)=1$,
$u'(\omega,0)=i\omega$ is derived in the form \[ u(\omega,x)=e^{i\omega
x}\left( 1+\frac{u_1(x)}{\omega}+ \frac{u_2(x)}{\omega^2}\right)
+\frac{e^{-i\omega x}u_3(x)}{\omega^2}-\frac{1}{\omega^2}\sum_{n=0}^{\infty}
i^{n}\alpha_n(x)j_n(\omega x), \] where $u_m(x)$, $m=1,2,3$ are given in a
closed form, $j_n$ stands for a spherical Bessel function of order $n$ and the
coefficients $\alpha_n$ are calculated by a recurrent integration procedure.
The following estimate is proved $\vert u(\omega,x) -u_N(\omega,x)\vert \leq
\frac{1}{\vert \omega \vert^2}\varepsilon_N(x)\sqrt{\frac{\sinh(2\mathop{\rm
Im}\omega\,x)}{\mathop{\rm Im}\omega}}$ for any $\omega\in\mathbb{C}\backslash
\{0\}$, where $u_N(\omega,x)$ is an approximate solution given by truncating
the series in the representation for $u(\omega,x)$ and $\varepsilon_N(x)$ is a
nonnegative function tending to zero for all $x$ belonging to a finite interval
of interest. In particular, for $\omega\in\mathbb{R}\backslash \{0\}$ the
estimate has the form $\vert u(\omega,x)-u_N(\omega,x)\vert \leq
\frac{1}{\vert\omega\vert^2}\varepsilon_N(x)$. A numerical illustration of
application of the new representation for computing the solution $u(\omega,x)$
on large sets of values of the spectral parameter $\omega$ with an accuracy
nondeteriorating (and even improving) when $\omega \rightarrow \pm \infty$ is
given.

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The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat