Abstract.
The eigenfunctions \( e^{i\langle\lambda,x\rangle} \) of the Laplacian on a flat torus have uniformly bounded L p norms. In this article, we prove that for every other quantum integrable Laplacian, the L p norms of the joint eigenfunctions blow up at least at the rate \( \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} \) when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in L p unless (M,g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.
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Submitted 13/06/02, accepted 5/02/03
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ID="*"Communicated by Bernard Helffer
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ID="**"Research partially supported by an Alfred P. Sloan Research Fellowship and NSERC grant No. OGP0170280
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ID="***"Research partially supported by NSF grant No. DMS-0071358
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Toth, J., Zelditch, S. L p Norms of Eigenfunctions in the Completely Integrable Case. Ann. Henri Poincaré 4, 343–368 (2003). https://doi.org/10.1007/s00023-003-0132-x
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DOI: https://doi.org/10.1007/s00023-003-0132-x