Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Schroedinger Operators on Regular Metric Trees with Long Range
  Potentials: Weak Coupling Behavior   Consider a regular $d$-dimensional metric tree $\Gamma$ with root $o$. Define
the Schroedinger operator $-\Delta - V$, where $V$ is a non-negative, symmetric
potential, on $\Gamma$, with Neumann boundary conditions at $o$. Provided that
$V$ decays like $x^{-\gamma}$ at infinity, where $1 < \gamma \leq d \leq 2,
\gamma \neq 2$, we will determine the weak coupling behavior of the bottom of
the spectrum of $-\Delta - V$. In other words, we will describe the
asymptotical behavior of $\inf \sigma(-\Delta - \alpha V)$ as $\alpha \to 0+$

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The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
27
The categories of this paper are: comp-gas