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Spreading of Lagrangian Regularity on Rational Invariant Tori

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Abstract

Let P h be a self-adjoint semiclassical pseudodifferential operator on a manifold M such that the bicharacteristic flow of the principal symbol on T*M is completely integrable and the subprincipal symbol of P h vanishes. Consider a semiclassical family of eigenfunctions, or, more generally, quasimodes u h of P h . We show that on a nondegenerate rational invariant torus, Lagrangian regularity of u h (regularity under test operators characteristic on the torus) propagates both along bicharacteristics, and also in an additional “diffractive” manner. In particular, in addition to propagating along null bicharacteristics, regularity fills in the interiors of small annular tubes of bicharacteristics.

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  1. Department of Mathematics, Northwestern University, 2033 Sheridan Rd., Evanston, IL, 60208, USA

    Jared Wunsch

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  1. Jared Wunsch
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Correspondence to Jared Wunsch.

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Communicated by P. Sarnak

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Wunsch, J. Spreading of Lagrangian Regularity on Rational Invariant Tori. Commun. Math. Phys. 279, 487–496 (2008). https://doi.org/10.1007/s00220-008-0434-4

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  • DOI: https://doi.org/10.1007/s00220-008-0434-4

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