Caustics of Weakly Lagrangian Distributions

Seán Gomes; Jared Wunsch

doi:10.1007/s00023-021-01110-8

Caustics of Weakly Lagrangian Distributions

Seán Gomes^*, Jared Wunsch

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study semiclassical sequences of distributions u_h associated with a Lagrangian submanifold of phase space L⊂ T^∗X. If u_h is a semiclassical Lagrangian distribution, which concentrates at a maximal rate on L, then the asymptotics of u_h 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 u_h 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)
Journal	Annales Henri Poincare
DOIs	https://doi.org/10.1007/s00023-021-01110-8
State	Accepted/In press - 2021

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-021-01110-8

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title = "Caustics of Weakly Lagrangian Distributions",

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{\textquoteright}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.",

author = "Se{\'a}n Gomes and Jared Wunsch",

note = "Funding Information: 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. ). St{\'e}phane 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. Publisher Copyright: {\textcopyright} 2021, Springer Nature Switzerland AG.",

year = "2021",

doi = "10.1007/s00023-021-01110-8",

language = "English (US)",

journal = "Annales Henri Poincare",

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AU - Gomes, Seán

AU - Wunsch, Jared

N1 - Funding Information: 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. Publisher Copyright: © 2021, Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

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Caustics of Weakly Lagrangian Distributions

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