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- W2897319785 abstract "Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2Delta_g+V(x)-E_1$ so that $dp_1neq 0$ 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 Hormander 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 Hormander 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 $xin M$ 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." @default.
- W2897319785 created "2018-10-26" @default.
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- W2897319785 date "2018-10-09" @default.
- W2897319785 modified "2023-10-16" @default.
- W2897319785 title "Pointwise bounds for joint eigenfunctions of quantum completely integrable systems" @default.
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