Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Balian-Low theorem for locally compact abelian groups and vector
  bundles   Let $\Lambda$ be a lattice in a second countable, locally compact abelian
group $G$ with annihilator $\Lambda^{\perp} \subseteq \widehat{G}$. We
investigate the validity of the following statement: For every $\eta$ in the
Feichtinger algebra $S_0(G)$, the Gabor system $\{ M_{\tau} T_{\lambda} \eta
\}_{\lambda \in \Lambda, \tau \in \Lambda^{\perp}}$ is not a frame for
$L^2(G)$. When $G = \mathbb{R}$ and $\Lambda = \alpha \mathbb{Z}$, this
statement is a variant of the Balian-Low theorem. Extending a result of R.
Balan, we show that whether the statement generalizes to $(G,\Lambda)$ is
equivalent to the nontriviality of a certain vector bundle over the compact
space $(G/\Lambda) \times (\widehat{G}/\Lambda^{\perp})$. We prove this
equivalence using a connection between Gabor frames and Heisenberg modules.
More specifically, we show that the Zak transform can be viewed as an
isomorphism of certain Hilbert $C^*$-modules. As an application, we prove a new
Balian-Low theorem for the group $\mathbb{R} \times \mathbb{Q}_p$, where
$\mathbb{Q}_p$ denotes the $p$-adic numbers.

28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph