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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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