Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fourth-order Schr\"odinger type operator with unbounded coefficients in
  $L^2(\mathbb{R}^N)$   In this paper we study generation results in $L^2(\mathbb{R}^N)$ for the
fourth order Schr\"odinger type operator with unbounded coefficients of the
form $$A=a^{2} \Delta ^2+V^{2}$$ where $a(x)=1+|x|^{\alpha}$ and
$V=|x|^{\beta}$ with $\alpha>0$ and $\beta >(\alpha-2)^+$. We obtain that
$(-A,D(A))$ generates an analytic strongly continuous semigroup in
$L^2(\mathbb{R}^N)$ for $N\geq5$. Moreover, the maximal domain $D(A)$ can be
characterized for $N>8$ by the weighted Sobolev space
  \[ D_2(A)=\{u\in H^{4}(\mathbb{R}^N)\,:\,V^{2}u\in L^{2}(\mathbb{R}^N),
|x|^{2\alpha-h}D^{4-h}u\in L^{2}(\mathbb{R}^N) \text{ for } h=0,1,2,3,4\}. \]

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The categories of this paper are: cmp-lg
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
4
The categories of this paper are: cond-mat
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
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The categories of this paper are: hep-th