Complex zeros of real ergodic eigenfunctions. (English) Zbl 1134.37005
Summary: We determine the limit distribution (as \(\lambda \rightarrow \infty\)) of complex zeros for holomorphic continuations \(\varphi_\lambda^{\mathbb C}\) to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold \((M, g)\) with ergodic geodesic flow. If \(\{\varphi_{j_k}\}\) is an ergodic sequence of eigenfunctions, we prove the weak limit formula \(\frac{1}{\lambda_j[Z_{\varphi_{j_k}^{\mathbb{C}}}]}\;\to\;\frac{i}{\pi} \partial\bar{\partial} |\xi|_g\), where \([Z_{\varphi_{j_k}^{\mathbb{C}}}]\) is the current of integration over the complex zeros and where \(\overline\partial\) is with respect to the adapted complex structure of L. Lempert and R. Szőke [Math. Ann. 290, No. 4, 689–712 (1991; Zbl 0752.32008)] and V. Guillemin and M. Stenzel [J. Differ. Geom. 34, No. 2, 561–570 (1991; Zbl 0746.32005)].
MSC:
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P05 | General topics in linear spectral theory for PDEs |
| 58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
| 53C20 | Global Riemannian geometry, including pinching |
| 53D25 | Geodesic flows in symplectic geometry and contact geometry |
Keywords:
Laplace equation; asymptotic distribution of complex modal hypersurfaces; quantum ergodic eigenfunctions; holomorphic continuations; Grauert tubes; Riemannian manifold; ergodic geodesic flowReferences:
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