Eigenfunctions of the Laplacian on compact Riemannian manifolds

Asian Journal of Mathematics

Volumn 10, Issue 1, 2006, Pages 115-126

  Donnelly, Harold ^a  

^a PURDUE UNIVERSITY   (United States)

Author keywords

Ergodic geodesic flow; Mathematical physics; Quantum chaos

Indexed keywords

EID: 43349102950     PISSN: 10936106     EISSN: 19450036     Source Type: Book Series    
DOI: 10.4310/AJM.2006.v10.n1.a7     Document Type: Article

References (25)

1. Aronszajn N., A unique continuation theorem for solutions of elliptic partial differential equa- tions, J. Math. Pures Appl., 36 (1957), 235-249.
2. Bishop, R. and Crittenden, R., Geometry of Manifolds, Academic Press, N, Y., 1964.
3. Bourgain, J. and Lindenstrauss, E., Entropy of quantum limits, Communications in Math- ematical Physics, 233 (2003), 153-171.
4. Bourgain, J., Eigenfunction bounds for compact manifolds with integrable geodesic flows, IHES preprint, 1993.
5. Braüning, J., Über Knoten von Eigenfunctionen des Laplace Beltrami Operator, Math. Z., 158 (1978), 15-21.
6. Burq, N. and Zworski, M., Geometric control in the presence of a black box, Journal of the American Mathematical Society, 17 (2004), 443-471.
7. Burns, K. and Gerber, M., Real analytic Bernoulli geodesic flows on S2, Ergodic theory and dynamical systems, 9 (1989), 27-45.
8. Colin De Verdière, Y., Ergodicite et functions propre du laplacian, Communications in Math- ematical Physics, 102 (1985), 497-502.
9. Cordes, H., Über die eindeutige Bestimmheit Lösungen elliptischer Differentialgleichungen durch anfangsvorgaben, Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II, 11 (1956), 239-258.
10. Dong, R.T., Nodal sets of eigenfunctions on Riemann surfaces, Journal of Differential Geom- etry, 36 (1992), 493-506.
11. Donnelly, H. and Fefferman, C., Nodal sets of eigenfunctions of the Laplacian on surfaces, Journal of the American Mathematical Society, 3 (1990), 333-353.
12. Donnelly, H. and Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Inventiones Math., 93 (1988), 161-183.
13. Donnelly, H., Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds, J. Func. Anal., 187 (2001), 247-261.
14. Donnelly, H., Quantum unique ergodicity, Proceedings of the American Mathematical Society, 131 (2003), 2945-2951.
15. Hardt, R. and Simon, L., Nodal sets for solution of elliptic equations, Journal of Differential Geometry, 30 (1989), 505-522.
16. Hörmander, L., The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218.
17. Lindenstrauss, E., Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics, to appear.
18. Rudnick, Z. and Sarnak, P., The behavior of eigenstates of arithmetic hyperbolic manifolds, Commun. Math. Phys., 161 (1994), 195-213.
19. Sarnak, P., Arithmetic quantum chaos, Blyth lecture series notes, University of Toronto, 1993.
20. Shnirelman, A.I., Ergodic properties of eigenfunctions, Uspehi Mat. Nauk, 29 (1974), 181-182.
21. Sogge, C., Concerning the Lp norm of spectral clusters of second order elliptic operators on compact manifolds, J. Func. Anal., 77 (1988), 123-168.
22. Toth, J.A. and Zelditch, S., Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J., 111 (2002), 97-132.
23. Zelditch, S., Uniform distribution of eigenfunctions on a compact hyperbolic surface, Duke Mathematics Journal, 55 (1987), 919-941.
24. Zelditch, S., Asymptotics of Polynomials and Eigenfunctions, Proceedings of the ICM 2002, Vol II, 733-742.
25. Zelditch, S., Note on quantum unique ergodicity, Proceedings of the American Mathematical Society, 132 (2004), 1869-1872.