Improvement of Pointwise Bounds for Eigenfunctions in the Quantum Completely Integrable System

On a compact n-dimensional Riemannian manifold without boundary (M,g), it is well-known that the L2-normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal L∞ growth bound of O(h1−n2)ash→0+. In the context of a quantum completely integrable system on M, which consists of n commuting self-adjoint pseudodifferential operators P1(h),…,Pn(h), where P1(h)=−h2Δg+V(x), Galkowski-Toth showed polynomial improvements over the standard O(h1−n2) bounds for typical points. Specifically, in the two-dimensional case, such an improved upper bound is O(h−1/4). In this study, we aim to further enhance this bound to O(|lnh|1/2) at the points where a strictly monotonic condition is satisfied.