Complex zeros of real ergodic eigenfunctions

By Steve Zelditch. 2005

48 citations recorded by the source corpus.

We determine the limit distribution (as λ→∞) of complex zeros for holomorphic continuations φλℂ to Grauert tubes of real eigenfunctions of the Laplacian on a real analytic compact Riemannian manifold (M,g) with ergodic geodesic flow. If $\{\phi_{j_{k}}\}$ is an ergodic sequence of eigenfunctions, we prove the weak limit formula $\frac{1}{\lambda_j}[Z_{\phi_{j_k}}^{\mathbb{C}}]\ \to\ \frac{i}{\pi} \partial\bar{\partial} |\xi|_g$, where $[Z_{\phi_{j_k}^{\mathbb{C}}}]$ is the current of integration over the complex zeros and where $\overline{\partial}$ is with respect to the adapted complex structure of Lempert-Szöke and Guillemin-Stenzel.

View Paper: https://www.semanticscholar.org/paper/fab08dddb2b8522c30fe7d9a4001b33b7809e60f

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Citations in the corpus

Counting Nodal Lines Which Touch the Boundary of an Analytic Domain (2007, methodology): “3.1. Complexification of domains and their boundaries ∂. When (M, g) is a real analytic Riemannian manifold without boundary, then there exists a complexification MC of M as an (open) complex manifold, and Laplace eigenfunctions extend to a maximal tube in MC. This has been studied in [Bou, GS1, GS2”
Real and complex zeros of Riemannian random waves (2008, background): “ ′(1,1)(B∗ εM). REAL AND COMPLEX ZEROS OF RIEMANNIAN RANDOM WAVES 5 As mentioned above, this result shows that the complex zeros of the random waves have the same expected limit distribution found in [Z3] for real analytic compact Riemannian manifolds with ergodic geodesic ﬂow. 0.3. The key objects ”
Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I (2011, background): “urier integral operator with a positive complex canonical relation. The real points of its canonical relation form the graph ∆Σ of the identity map on the symplectic cone Στ ⊂ T∗∂Mτ (51). We refer to [Z3] for further background. We only need the ﬁrst statement in the following: Lemma 8.2. Let Ψs(X) ”