Caustics of weakly Lagrangian distributions. (English) Zbl 1487.35008
Summary: We study semiclassical sequences of distributions \(u_h\) associated with a Lagrangian submanifold of phase space \(\mathcal{L}\subset T^*X\). If \(u_h\) is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on \(\mathcal{L},\) then the asymptotics of \(u_h\) are well understood by work of Arnol’d, provided \(\mathcal{L}\) projects to \(X\) with a stable simple Lagrangian singularity. We establish sup-norm estimates on \(u_h\) under much more general hypotheses on the rate at which it is concentrating on \(\mathcal{L}\) (again assuming a stable simple projection). These estimates apply to sequences of eigenfunctions of integrable and KAM Hamiltonians.
MSC:
| 35A08 | Fundamental solutions to PDEs |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35P05 | General topics in linear spectral theory for PDEs |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 53D12 | Lagrangian submanifolds; Maslov index |
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