Galkowski, J; Toth, JA; (2020) Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems. Communications in Mathematical Physics , 375 pp. 915-947. 10.1007/s00220-020-03730-3.

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Abstract

Let (M, g) be a compact Riemannian manifold of dimension n and P1:=−h2Δg+V(x)−E1 so that dp1≠0 on p1=0. We assume that P1 is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators P2,…Pn with [Pi,Pj]=0, i,j=1,…n. We study the pointwise bounds for the joint eigenfunctions, uh of the system {Pi}ni=1 with P1uh=E1uh+o(1). In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M. In two and three dimensions, these estimates agree with the Hardy exponent h−1−n4 and in higher dimensions we obtain a gain of h12 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 x∈M in the “microlocally forbidden” region p−11(E1)∩⋯∩p−1n(En)∩T∗xM=∅. These bounds are sharp locally near the projection of the invariant tori.

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