Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pointwise bounds for joint eigenfunctions of quantum completely
  integrable systems   Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2\Delta_g+V(x)-E_1$
so that $dp_1\neq 0$ on $p_1=0$. We assume that $P_1$ is quantum completely
integrable in the sense that there exist functionally independent
pseuodifferential operators $P_2,\dots P_n$ with $[P_i,P_j]=0$, $i,j=1,\dots
,n$. We study the pointwise bounds for the joint eigenfunctions, $u_h$ of the
system $\{P_i\}_{i=1}^n$ with $P_1u_h=E_1u_h+o(1)$. We first give polynomial
improvements over the standard H\"ormander bounds for typical points in $M$. In
two and three dimensions, these estimates agree with the Hardy exponent
$h^{-\frac{1-n}{4}}$ and in higher dimensions we obtain a gain of
$h^{\frac{1}{2}}$ over the H\"ormander bound.
  In our second main result, under a real-analyticity assumption on the QCI
system, we give exponential decay estimates for joint eigenfunctions at points
outside the projection of invariant Lagrangian tori; that is at points $x\in M$
in the "microlocally forbidden" region $p_1^{-1}(E_1)\cap \dots \cap
p_n^{-1}(E_n)\cap T^*_xM=\emptyset.$ These bounds are sharp locally near the
projection of the invariant tori.

28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs