Abstract
It is an open problem in general to prove that there exists a sequence of (Formula presented.) -eigenfunctions (Formula presented.) on a Riemannian manifold (M, g) for which the number (Formula presented.) of nodal domains tends to infinity with the eigenvalue. Our main result is that (Formula presented.) along a subsequence of eigenvalues of density 1 if (M, g) is a non-positively curved surface with concave boundary, i.e., a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 813-840 |
| Number of pages | 28 |
| Journal | Mathematische Annalen |
| Volume | 364 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - Apr 1 2016 |
Funding
This paper makes use of recent joint work of the second author with several collaborators, which in part was motivated by the applications to nodal sets. Theorem is recent joint work with X. Han, A. Hassell and H. Hezari []. It also uses calculations in recent work [] with H. Hezari.As mentioned above, Theorem is joint work with C. D. Sogge []. The boundary quantum ergodicity theorem and boundary local Weyl law is joint work with A. Hassell [] and with H. Christianson and J. Toth []. We would also like to thank N. Simanyi for helpful correspondence regarding [] and L. Stoyanov for correspondence on billiard problems. The first author was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIP) (No. 2013042157). The first author was also partially supported by NSF grant DMS-1128155 and by TJ Park Post-doc Fellowship funded by POSCO TJ Park Foundation. Research of the second author was partially supported by NSF grant DMS-1206527.
ASJC Scopus subject areas
- General Mathematics