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

Xianchao Wu ()
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Xianchao Wu: School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430070, China

Mathematics, 2025, vol. 13, issue 17, 1-9

Abstract: On a compact n -dimensional Riemannian manifold without boundary ( M , g ) , it is well-known that the L 2 -normalized Laplace eigenfunctions with semiclassical parameter h satisfy the universal L ∞ growth bound of O ( h 1 − n 2 ) as h → 0 + . In the context of a quantum completely integrable system on M , which consists of n commuting self-adjoint pseudodifferential operators P 1 ( h ) , … , P n ( h ) , where P 1 ( h ) = − h 2 Δ g + V ( x ) , Galkowski-Toth showed polynomial improvements over the standard O ( h 1 − n 2 ) 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 ( | ln h | 1 / 2 ) at the points where a strictly monotonic condition is satisfied.

Keywords: quantum completely integrable system; eigenfunction; stationary phase (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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