Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sufficient conditions for two-dimensional localization by arbitrarily
  weak defects in periodic potentials with band gaps   We prove, via an elementary variational method, 1d and 2d localization within
the band gaps of a periodic Schrodinger operator for any mostly negative or
mostly positive defect potential, V, whose depth is not too great compared to
the size of the gap. In a similar way, we also prove sufficient conditions for
1d and 2d localization below the ground state of such an operator. Furthermore,
we extend our results to 1d and 2d localization in d dimensions; for example, a
linear or planar defect in a 3d crystal. For the case of D-fold degenerate band
edges, we also give sufficient conditions for localization of up to D states.

2
The categories of this paper are: math
0
The categories of this paper are: hep-ph
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
19
The categories of this paper are: eess
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
1
The categories of this paper are: cs
31
The categories of this paper are: ao-sci