Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Low-lying eigenvalues of semiclassical Schr\"odinger operator with
  degenerate wells   In this article, we consider the semiclassical Schr\"odinger operator $P = -
h^{2} \Delta + V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$
which vanishes, and study its low-lying eigenvalues $\lambda_{k} ( P )$ as $h
\to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$
for $\lambda_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) =
\{ 0 \}$, we are able to control the eigenvalues $\lambda_{k} ( P )$ for
monotonous potentials by a quantity linked to an interval $I_{h}$, determined
by an implicit relation involving $V$ and $h$. Next, we consider the case where
$V$ has a flat minimum, in the sense that it vanishes to infinite order. We
give the asymptotic of the eigenvalues: they behave as the eigenvalues of the
Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the
associated eigenvectors and extends in particular cases to higher dimensions.

13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics