Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral estimates for the Schr\"odinger operators with sparse
  potentials on graphs   The construction of "sparse potentials", suggested in \cite{RS09} for the
lattice $\Z^d,\ d>2$, is extended to a wide class of combinatorial and metric
graphs whose global dimension is a number $D>2$. For the Schr\"odinger operator
$-\D-\a V$ on such graphs, with a sparse potential $V$, we study the behavior
(as $\a\to\infty$) of the number $N_-(-\D-\a V)$ of negative eigenvalues of
$-\D-\a V$. We show that by means of sparse potentials one can realize any
prescribed asymptotic behavior of $N_-(-\D-\a V)$ under very mild regularity
assumptions. A similar construction works also for the lattice $\Z^2$, where
D=2.

35
The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
17
The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
3
The categories of this paper are: physics
19
The categories of this paper are: eess
29
The categories of this paper are: funct-an