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$$L^p$$ norms of eigenfunctions in the completely integrable case. (English) Zbl 1028.58028
Summary: 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 $$\|\varphi_k \|L^p \geq C(\varepsilon)\lambda_{k}^{\frac{p-2}{4p}-\varepsilon}$$ when $$p>2$$. This gives a quantitative refinement of our recent result [Duke Math. J. 111, 97-132 (2002; Zbl 1022.58013)] 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.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 35P20 Asymptotic distributions of eigenvalues in context of PDEs 37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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