Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Reducibility of 1-D Quantum Harmonic Oscillator with Decaying Conditions
  on the Derivative of Perturbation Potentials   We prove the reducibility of 1-D quantum harmonic oscillators in $\mathbb R$
perturbed by a quasi-periodic in time potential $V(x,\omega t)$ under the
following conditions, namely there is a $C>0$ such that \begin{equation*}
|V(x,\theta)|\le C,\quad|x\partial_xV(x,\theta)|\le
C,\quad\forall~(x,\theta)\in\mathbb R\times\mathbb T_\sigma^n. \end{equation*}
The corresponding perturbation matrix $(P_i^j(\theta))$ is proved to satisfy
$(1+|i-j|)| P_i^j(\theta)|\le C$ and
$\sqrt{ij}|P_{i+1}^{j+1}(\theta)-P_i^j(\theta)|\le C$ for any $\theta\in\mathbb
T_\sigma^n$ and $i,j\geq 1$. A new reducibility theorem is set up under this
kind of decay in the perturbation matrix element $P_{i}^j(\theta)$ as well as
the discrete difference matrix element $P_{i+1}^{j+1}(\theta)-P_i^j(\theta)$.
For the proof the novelty is that we use the decay in the discrete difference
matrix element to control the measure estimates for the thrown parameter sets.

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The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
31
The categories of this paper are: ao-sci