Riemannian manifolds with maximal eigenfunction growth

Duke Mathematical Journal

Volumn 114, Issue 3, 2002, Pages 387-437

  Sogge, Christopher D ^a   Zelditch, Steve ^a  

^a JOHNS HOPKINS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037106707     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-02-11431-8     Document Type: Article

References (40)

1. [A] V.I. Arnold, Mathematical Methods of Classical Mechanics, 2d ed., Grad. Texts in Math. 60, Springer, New York, 1998. MR 96c:70001.
2. [Av] V.G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956), 327-344. MR 18:316a.
3. [Be] P.H. Bérard, On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z. 155 (1977), 249-276. MR 56:13295.
4. [BB] L. Bérard-Bergery, Quelques exemples de variétés riemanniennes où toutes les géodésiques issues d'un point sont fermées et de même longueur, suivis de quelques résultats sur leur topologie, Ann. Inst. Fourier (Grenoble) 27 (1977), 231-249. MR 55:9124.
5. [Bes] A.L. Besse, Manifolds All of Whose Geodesics Are Closed, with appendices by D.B.A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger, and J.L. Kazdan, Ergeb. Math. Grenzgeb. 93, Springer, Berlin, 1978. MR 80c:53044.
6. [B1] P.M. Bleher, Distribution of energy levels of a quantum free particle on a surface of revolution, Duke Math J. 74 (1994), 45 -93. MR 96f:58169.
7. [B] J. Bourgain, Eigenfunction bounds for compact manifolds with integrable geodesic flow, preprint, Institut des Hantes Études Scientifiqnes, 1993.
8. [CV] Y. Colin De Verdière, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II: Le cas intégrable, Math. Z. 171 (1980), 51-73. MR 8li:58046.
9. [CVV] Y. Colin De Verdière and S. Vũ ngoc, Singular Bohr-Sommerfeld rules for 2D integrable systems, preprint, 2000, http://www-fourier.ujf-grenoble.fr/PREP/html/a508/a508.html.
10. [D] J.J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687-706. MR 82d:58029.
11. [DG] J.J. Duistermaat and V.W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79. MR 53:9307.
12. [H] R.M. Hardt, "Stratification via corank one projections" in Singularities (Arcata, Calif., 1981), Part 1, Proc. Sympos. Pure Math. 40, Amer. Math. Soc., Providence, 1983, 559-566. MR 85e:32016.
13. [Ho1] L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. MR 58:29418.
14. [Ho2] _, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985. MR 87d:35002a.
15. [Ho3] _, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Gmndlehren Math. Wiss. 275, Springer, Berlin, 1985. MR 87d:35002b.
16. [Iv1] V.Ja. Ivriǐ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary (in Russian), Funktsional. Anal. i Prilozhen. 14, no. 2 (1980), 25-34. MR 82m:58057.
17. [Iv2] _, "Semiclassical spectral asymptotics and multiparticle quantum theory" in Partial Differential Operators and Mathematical Physics (Holzhau, Germany, 1994), Oper. Theory Adv. Appl. 78, Birkhäuser, Basel, 1995, 199-212. MR 97e:81177.
18. [Ka] A. Karnaukh, Spectral count on compact negatively curved surfaces, Ph.D. dissertation, Princeton University, Princeton, 1995.
19. [K] W. Klingenberg, Riemannian Geometry, de Gruyter Stud. Math. 1, de Gruyter, Berlin, 1982. MR 84j:53001.
20. [KMS] D.V. Kosygin, A.A. Minasov, and Ya.G. Sinaǐ, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces (in Russian), Uspekhi Mat. Nauk 48, no. 4 (292) (1993), 3-130; English translation in Russian Math. Surveys 48, no. 4 (1993), 1-142. MR 95e:58174.
21. [KMS] D.V. Kosygin, A.A. Minasov, and Ya.G. Sinaǐ, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces (in Russian), Uspekhi Mat. Nauk 48, no. 4 (292) (1993), 3-130; English translation in Russian Math. Surveys 48, no. 4 (1993), 1-142. MR 95e:58174.
22. [L] B.M. Levitan, On the asymptoptic behavior of the spectral function of a self-adjoint differential equation of the second order, Izv. Akad. Nank SSSR Set. Mat. 16 (1952), 325-352. MR 15:315e.
23. [M] J. Milnor, Morse Theory, based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud. 51, Princeton Univ. Press, Princeton, 1963. MR 29:634.
24. [N] R. Narasimhan, Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25, Springer, Berlin, 1966. MR 36:428.
25. [S] Yu.G. Safarov, Asymptotics ora spectral function ora positive elliptic operator without a nontrapping condition (in Russian), Funktsional. Anal. i Prilozhen 22, no. 3 (1988), 53-65; English translation in Funct. Anal. Appl. 22 (1988), 213-223. MR 89i:35108.
26. [S] Yu.G. Safarov, Asymptotics ora spectral function ora positive elliptic operator without a nontrapping condition (in Russian), Funktsional. Anal. i Prilozhen 22, no. 3 (1988), 53-65; English translation in Funct. Anal. Appl. 22 (1988), 213-223. MR 89i:35108.
27. [SV] Yu. Safarov and D. Vassiliev, The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Transl. Math. Monogr. 155, Amer. Math. Soc., Providence, 1997. MR 97h:58163.
28. [Sa] P. Sarnak, in preparation.
29. [So1] C.D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J. 53 (1986), 43-65. MR 87g:42026.
30. p norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123-138. MR 89d:35131.
31. [So3] _, Fourier Integrals in Classical Analysis, Cambridge Tracts in Math. 105, Cambridge Univ. Press, Cambridge, 1993. MR 94c:35178.
32. [T] M.E. Taylor, The Gibbs phenomenon, the Pinsky phenomenon, and variants for eigenfunction expansions, Comm. Partial Differential Equations 27 (2002), 565-605.
33. [TZ1] J.A. Toth AND S. Zelditch, Riemannian manifolds with uniformly bounded eigenfunctions, Duke Math. J. 111 (2002), 97-132. CMP 1 876 442.
34. p-norms of eigenfunctions in the completely integrable case, preprint, 2001.
35. ∞ norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices 1997, 329-347, MR 99d:58175; Correction, Internat. Math. Res. Notices 1998, 65. MR 99d:58176.
36. ∞ norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices 1997, 329-347, MR 99d:58175; Correction, Internat. Math. Res. Notices 1998, 65. MR 99d:58176.
37. [Vo] A.V. Volovoy, Improved two-term asymptotics for the eigenvalue distribution function of an elliptic operator on a compact manifold, Comm. Partial Differential Equations 15 (1990), 1509-1563. MR 91m:58158.
38. [Z1] S. Zelditch, On the generic spectrum of a Riemannian cover, Ann. Inst. Fourier (Grenoble) 40 (1990), 407-442. MR 91g:58294.
39. [Z2] _, Kuznecov sum formulae and Szegö limit formulae on manifolds, Comm. Partial Differential Equations 17 (1992), 221-260. MR 93e:58184.
40. [Z3] _, The inverse spectral MR problem for surfaces of revolution, J. Differential Geom. 49 (1998), 207-264. MR 99k:58188.