Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hill's Spectral Curves and the Invariant Measure of the Periodic KdV
  Equation   This paper analyses the periodic spectrum of Schr\"odinger's equation
$-f''+qf=\lambda f$ when the potential is real, periodic, random and subject to
the invariant measure $\nu_N^\beta$ of the periodic KdV equation. This
$\nu_N^\beta$ is the modified canonical ensemble, as given by Bourgain ({Comm.
Math. Phys.} {166} (1994), 1--26), and $\nu_N^\beta$ satisfies a logarithmic
Sobolev inequality. Associated concentration inequalities control the
fluctuations of the periodic eigenvalues $(\lambda_n)$. For $\beta, N>0$ small,
there exists a set of positive $\nu_N^\beta$ measure such that $(\pm
\sqrt{2(\lambda_{2n}+\lambda_{2n-1})})_{n=0}^\infty$ gives a sampling sequence
for Paley--Wiener space $PW(\pi )$ and the reproducing kernels give a Riesz
basis. Let $(\mu_j)_{j=1}^\infty$ be the tied spectrum; then
$(2\sqrt{\mu_j}-j)$ belongs to a Hilbert cube in $\ell^2$ and is distributed
according to a measure that satisfies Gaussian concentration for Lipschitz
functions. The sampling sequence $(\sqrt{\mu_j})_{j=1}^\infty$ arises from a
divisor on the spectral curve, which is hyperelliptic of infinite genus. The
linear statistics $\sum_j g(\sqrt{\lambda_{2j}})$ with test function $g\in
PW(\pi)$ satisfy Gaussian concentration inequalities.

36
The categories of this paper are: cmp-lg
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg