Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The spectrum of Schr\"odinger operators and Hodge Laplacians on
  conformally cusp manifolds   We describe the spectrum of the $k$-form Laplacian on conformally cusp
Riemannian manifolds. The essential spectrum is shown to vanish precisely when
the $k$ and $k-1$ de Rham cohomology groups of the boundary vanish. We give
Weyl-type asymptotics for the eigenvalue-counting function in the purely
discrete case. In the other case we analyze the essential spectrum via positive
commutator methods and establish a limiting absorption principle. This implies
the absence of the singular spectrum for a wide class of metrics. We also
exhibit a class of potentials $V$ such that the Schr\"odinger operator has
compact resolvent, although $V$ tends to $-\infty$ in most of the infinity. We
correct a statement from the literature regarding the essential spectrum of the
Laplacian on forms on hyperbolic manifolds of finite volume, and we propose a
conjecture about the existence of such manifolds in dimension four whose cusps
are rational homology spheres.

32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
21
The categories of this paper are: adap-org