Galkowski Jeffrey, Toth John A
Department of Mathematics, University College London, London, UK.
Department of Mathematics and Statistics, McGill University, Montréal, QC Canada.
Commun Math Phys. 2020;375(2):915-947. doi: 10.1007/s00220-020-03730-3. Epub 2020 Apr 5.
Let (, ) be a compact Riemannian manifold of dimension and so that on . We assume that is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators with , . We study the pointwise bounds for the joint eigenfunctions, of the system with . In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in . In two and three dimensions, these estimates agree with the Hardy exponent and in higher dimensions we obtain a gain of over the Hörmander bound. In our second main result (Theorem 3), 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 in the "microlocally forbidden" region These bounds are sharp locally near the projection of the invariant tori.
设((M, g))是一个(n)维紧致黎曼流形，使得在(M)上(g)满足某种条件。我们假设(M)在量子完全可积（ACI）的意义下，即存在函数独立的拟微分算子(P_1, \cdots, P_n)，其主象征为(p_1, \cdots, p_n)。我们研究该系统(P_1, \cdots, P_n)的联合本征函数(\psi)的逐点界。在定理1中，我们首先针对(M)中的典型点给出了比标准霍尔曼德界更好的多项式改进。在二维和三维中，这些估计与哈代指数一致，在更高维中我们相对于霍尔曼德界获得了(1)的增益。在我们的第二个主要结果（定理3）中，在对QCI系统的实解析性假设下，我们给出了不变拉格朗日环面投影之外的点处联合本征函数的指数衰减估计；即在“微局部禁止”区域(\Omega)中的点(x)处。这些界在不变环面投影附近局部是精确的。
