Partially localized quasimodes in large subspaces
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- 作者：Shimon Brooks
- 刊名：Israel Journal of Mathematics
- 出版年：2013
- 出版时间：November 2013
- 年：2013
- 卷：198
- 期：1
- 页码：393-417
- 全文大小：330KB
- 参考文献：1. N. Anantharaman and S. Nonnenmacher, / Entropy of semiclassical measures of the Walsh-quantized baker’s map, Annales Henri Poincaré 8 (2007), 37-4. CrossRef
2. P. H. Bérard, / On the wave equation on a compact Riemannian manifold without conjugate points, Mathematische Zeitschrift 155 (1977), 249-76. CrossRef
3. S. Brooks and E. Lindenstrauss, / Joint quasimodes, positive entropy, and quantum unique ergodicity, 2012, submitted, arXiv:1112.5311.
4. F. Faure, S. Nonnenmacher and S. De Bièvre, / Scarred eigenstates for quantum cat maps of minimal periods, Communications in Mathematical Physics 239 (2003), 449-92. CrossRef
5. S. Helgason, / Topics in Harmonic Analysis on Homogeneous Spaces, Progress in Mathematics, Vol. 13, Birkh?user Boston, Mass., 1981.
6. H. Iwaniec and P. Sarnak, / L ?/sup> / norms of eigenfunctions of arithmetic surfaces, The Annals of Mathematics 141 (1995), 301-20. CrossRef
7. H. Iwaniec, / Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, second edition, Vol. 53, American Mathematical Society, Providence, RI, 2002.
8. D. Kelmer, / Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms Communications in Mathematical Physics 276 (2007), 381-95. CrossRef
9. E. Lindenstrauss, / On quantum unique ergodicity for Γ\?×?, International Mathematics Research Notices 17 (2001), 913-33. CrossRef
10. E. Lindenstrauss, / Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics 163 (2006), 165-19. CrossRef
11. Z. Rudnick and P. Sarnak, / The behaviour of eigenstates of arithmetic hyperbolic manifolds, Communications in Mathematical Physics 161 (1994), 195-13. CrossRef
12. P. Sarnak, / Letter to morawetz, http://www.math.princeton.edu/sarnak/sarnaklettertomorawetz.pdf.
13. P. Sarnak, / Statistical properties of eigenvalues of the Hecke operators, in / Analytic Number Theory and Diophantine Problems (Stillwater, OK, 1984), Progress in Mathematics, Vol. 70, Birkh?user Boston, Boston, MA, 1987, pp. 321-31. CrossRef
14. P. Sarnak, / Recent progress on the quantum unique ergodicity conjecture, Bulletin of the American Mathematical Society (N.S.) 48 (2011), 211-28. CrossRef
15. S. Zelditch, / Pseudodifferential analysis on hyperbolic surfaces, Journal of Functional Analysis 68 (1986), 72-05. CrossRef
16. S. Zelditch, / Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Mathematical Journal 55 (1987), 919-41. CrossRef - 作者单位：Shimon Brooks (1)
1. Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, 11794-3660, USA - ISSN：1565-8511
文摘
We consider spaces of high-energy quasimodes for the Laplacian on a compact hyperbolic surface, and show that when the spaces are large enough, one can find quasimodes that exhibit strong localization phenomena. Namely, take any constant c, and a sequence of cr j -dimensional spaces S j of quasimodes, where $\frac{1}{4} + r_j^2 \to \infty $ is an approximate eigenvalue for S j . Then we can find a sequence of vectors ψ j ?S j , such that any weak-* limit point of the microlocal lifts of |ψ j |2 localizes a positive proportion of its mass on a singular set of codimension 1. This result is sharp, in light of the QUE result of [BL12] for certain joint quasimodes that include spaces of size o(r j ), with arbitrarily slow decay.