Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Three circles theorems for Schrodinger operators on cylindrical ends and
  geometric applications   We show that for a Schr\"odinger operator with bounded potential on a
manifold with cylindrical ends the space of solutions which grows at most
exponentially at infinity is finite dimensional and, for a dense set of
potentials (or, equivalently for a surface, for a fixed potential and a dense
set of metrics), the constant function zero is the only solution that vanishes
at infinity. Clearly, for general potentials there can be many solutions that
vanish at infinity.
  These results follow from a three circles inequality (or log convexity
inequality) for the Sobolev norm of a solution $u$ to a Schr\"odinger equation
on a product $N\times [0,T]$, where $N$ is a closed manifold with a certain
spectral gap. Examples of such $N$'s are all (round) spheres $\SS^n$ for $n\geq
1$ and all Zoll surfaces.
  Finally, we discuss some examples arising in geometry of such manifolds and
Schr\"odinger operators.

37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
17
The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat