Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dispersive estimates for four dimensional Schr\"{o}dinger and wave
  equations with obstructions at zero energy   We investigate $L^1(\mathbb R^4)\to L^\infty(\mathbb R^4)$ dispersive
estimates for the Schr\"odinger operator $H=-\Delta+V$ when there are
obstructions, a resonance or an eigenvalue, at zero energy. In particular, we
show that if there is a resonance or an eigenvalue at zero energy then there is
a time dependent, finite rank operator $F_t$ satisfying $\|F_t\|_{L^1\to
L^\infty} \lesssim 1/\log t$ for $t>2$ such that
$$\|e^{itH}P_{ac}-F_t\|_{L^1\to L^\infty} \lesssim t^{-1},\,\,\,\,\,\text{for}
t>2.$$ We also show that the operator $F_t=0$ if there is an eigenvalue but no
resonance at zero energy. We then develop analogous dispersive estimates for
the solution operator to the four dimensional wave equation with potential.

11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph