Abstract
We study semiclassical sequences of distributions uh associated with a Lagrangian submanifold of phase space L⊂ T∗X. If uh is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on L, then the asymptotics of uh are well understood by work of Arnol’d, provided L projects to X with a stable simple Lagrangian singularity. We establish sup-norm estimates on uh under much more general hypotheses on the rate at which it is concentrating on L (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1205-1237 |
| Number of pages | 33 |
| Journal | Annales Henri Poincare |
| Volume | 23 |
| Issue number | 4 |
| DOIs | |
| State | Published - Apr 2022 |
Funding
The authors are grateful to Steve Zelditch for helpful discussions and to Ilya Khayutin for explaining the number-theoretic literature on lattice point counting in shrinking spherical caps (Sect. ). Stéphane Nonnenmacher as well as two anonymous referees made many helpful suggestions on the exposition; one of the latter pointed out an error in the inductive step proving the main theorem. JW gratefully acknowledges partial support from Simons Foundation Grant 631302 and from NSF Grant DMS–1600023.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics