Ergodicity and intersections of nodal sets and geodesics on real analytic surfaces.(English)Zbl 1303.32002

This paper is part of a program of which the ultimate objective is to establish that the nodal sets of the eigenfunctions $$\varphi_j$$ of the Laplacian on a real-analytic Riemannian surface, with ergodic geodesic flow on the tangent bundle, become equidistributed, in an appropriate sense, as $$j$$ tends to infinity.
In this paper, the main result is that “in the complex domain, as the eigenvalue goes to $$\infty$$, the intersections of nodal sets with generic periodic and non-periodic geodesics condense along the underlying real geodesic and become uniformly distributed with respect to its arclength,” along a subsequence $$(\lambda_{j_k})$$ of eigenvalues of density one. The results involve an asymmetry condition spelled out in an appendix, in which material from J. A. Toth and S. Zelditch [Geom. Funct. Anal. 23, No. 2, 715–775 (2013; Zbl 1277.53088)] is recalled. The second result “is somewhat weaker due to the non-compactness of non-periodic geodesics.” The proofs rely on the calculus of Toeplitz operators developed by Boutet de Monvel and Guillemin.

MSC:

 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 58J51 Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity 58J40 Pseudodifferential and Fourier integral operators on manifolds

Zbl 1277.53088
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