Abstract
Let (M, g) be a compact Riemannian manifold of dimension n and P_1:=-h^2Delta _g+V(x)-E_1 so that dp_1ne 0 on p_1=0. We assume that P_1 is quantum completely integrable (ACI) 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). 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^{-frac{1-n}{4}} and in higher dimensions we obtain a gain of h^{frac{1}{2}} 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 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.
Highlights
Let (Mn, g) be a closed, compact C∞ manifold and P1(h) : C∞(M) → C∞(M) a self-adjoint semiclassical pseudodifferential operator of order m that is elliptic in the classical sense, i.e. | p1(x, ξ )| ≥ c|ξ |m − C
In that case we say that P1(h) is quantum completely integrable (QCI)
Given the joint eigenvalues E(h) = (E1(h), . . . , En(h)) ∈ Rn of P1(h), . . . Pn(h) we denote an L2normalized joint eigenfunction with joint eigenvalue E(h) by u E,h and J
Summary
Let (Mn, g) be a closed, compact C∞ manifold and P1(h) : C∞(M) → C∞(M) a self-adjoint semiclassical pseudodifferential operator of order m that is elliptic in the classical sense, i.e. | p1(x, ξ )| ≥ c|ξ |m − C. Under the real-analyticity assumption the decay estimate in Theorem 3 is sharp and improves on results of the second author in [Tot98]. In the cases where there exist appropriate coordinates in terms of which the classical generating function is separable, one can show that the decay estimates in Theorem 3 are still satisfied for non-generic energy levels E ∈ Breg. The latter condition is satisfied in all cases that we know of (see remark 3.5 for more details)
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