Riemannian manifolds with uniformly bounded eigenfunctions



Duke Mathematical Journal

Riemannian manifolds with uniformly bounded eigenfunctions

John A. Toth and Steve Zelditch

Source: Duke Math. J. Volume 111, Number 1 (2002), 97-132.

Abstract

The standard eigenfunctions $\phi_\lambda=e^{i\langle\lambda,x\rangle}$ on flat tori $\mathbb {R}^n/L$ have $L^\infty$-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that $L^2$-normalized eigenfunctions have uniformly bounded $^\infty$-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians.

Primary Subjects: 58J50
Secondary Subjects: 53D25

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.dmj/1087575008
Mathematical Reviews number (MathSciNet): MR1876442
Digital Object Identifier: doi:10.1215/S0012-7094-02-11113-2
Zentralblatt MATH identifier: 1022.58013

References

[AM] R. Abraham and J. E. Marsden, Foundations of Mechanics, 2d ed., Benjamin/Cummings, Reading, Mass., 1978.
Mathematical Reviews (MathSciNet): MR81e:58025
Zentralblatt MATH: 0393.70001
[A] V. I. Arnold, Modes and quasimodes (in Russian), Funktsional. Anal. i Prilo\v zen. 6, no. 2 (1972), 12--20.; English translation in Funct. Anal. Appl. 6 (1972), 94--101.
Mathematical Reviews (MathSciNet): MR45:6331
[Be1] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), 2083--2091.
Mathematical Reviews (MathSciNet): MR58:8961
[Be2] —, Semi-classical mechanics in phase space: A study of Wigner's function, Philos. Trans. Roy. Soc. London Ser. A 287 (1977), 237--271.
Mathematical Reviews (MathSciNet): MR58:8886
[BP] M. Bialy and L. Polterovich, Hopf-type rigidity for Newton equations, Math. Res. Lett. 2 (1995), 695--700.
Mathematical Reviews (MathSciNet): MR96m:58077
[BKS] P. Bleher, D. Kosygin, and Ya. G. Sinaĭ, Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae, Comm. Math. Phys. 179 (1995), 375--403.
Mathematical Reviews (MathSciNet): MR96c:58173
[BI] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), 259--269.
Mathematical Reviews (MathSciNet): MR95h:53049
[Ch] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent, Asympt. Anal. 1 (1988), 227--261.
Mathematical Reviews (MathSciNet): MR89j:35100
[CV1] Y. Colin de Verdière, Quasi-modes sur les variétés riemanniennes, Invent. Math. 43 (1977), 15--52.
Mathematical Reviews (MathSciNet): MR58:18615
[CV2] —, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques, Comment. Math. Helv. 54 (1979), 508--522.
Mathematical Reviews (MathSciNet): MR81a:58052
[11]\bibitem [CV3]CV1 —, Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, II: Le cas intégrable, Math. Z. 171 (1980), 51--73.
Mathematical Reviews (MathSciNet): MR81i:58046
[CP] Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique, I: Concentration microlocale, Comm. Partial Differential Equations 19 (1994), 1535--1563.
Mathematical Reviews (MathSciNet): MR96b:58112
[CK] C. Croke and B. Kleiner, On tori without conjugate points, Invent. Math. 120 (1995), 241--257.
Mathematical Reviews (MathSciNet): MR96j:53037
[D] J. J. Duistermaat, On global action-angle coordinates, Comm. Pure Appl. Math. 33 (1980), 687--706.
Mathematical Reviews (MathSciNet): MR82d:58029
[E] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein, Verh. Deutsch. Phys. Ges. 19 (1917), 82--92.
[GS1] V. Guillemin and S. Sternberg, Geometric Asymptotics, Math. Surveys 14, Amer. Math. Soc., Providence, 1977.
Mathematical Reviews (MathSciNet): MR58:24404
Zentralblatt MATH: 0364.53011
[GS2] —, Homogeneous quantization and multiplicities of group representations, J. Funct. Anal. 47 (1982), 344--380.
Mathematical Reviews (MathSciNet): MR84d:58034
[He] G. J. Heckman, Quantum integrability for the Kovalevsky top, Indag. Math. (N.S.) 9 (1998), 359--365.
Mathematical Reviews (MathSciNet): MR2000c:37082
[H] B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications, Lecture Notes in Math. 1336, Springer, Berlin, 1988.
Mathematical Reviews (MathSciNet): MR90c:81043
[Ho] L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren. Math. Wiss. 275, Springer, Berlin, 1985.
Mathematical Reviews (MathSciNet): MR87d:35002b
[JZ] D. Jakobson and S. Zelditch, ``Classical limits of eigenfunctions for some completely integrable systems'' in Emerging Applications of Number Theory (Minneapolis, 1996), IMA Vol. Math. Appl. 109, Springer, New York, 1999.
Mathematical Reviews (MathSciNet): MR2001g:58053
Zentralblatt MATH: 01552111
[K] A. Knauf, Closed orbits and converse KAM theory, Nonlinearity 3 (1990), 961--973.
Mathematical Reviews (MathSciNet): MR91f:58075
Digital Object Identifier: doi:10.1088/0951-7715/3/3/019
[KMS] D. Kosygin, A. Minasov, and Ya. G. Sinaĭ, Statistical properties of the spectra of Laplace-Beltrami operators on Liouville surfaces (in Russian), Uspekhi Mat. Nauk 48, no. 4 (1993), 3--130.; English translation in Russian Math. Surveys 48, no. 4 (1993), 1--142.
Mathematical Reviews (MathSciNet): MR95e:58174
[LaS] F. Lalonde and J.-C. Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents, Comment. Math. Helv. 66 (1991), 18--33.
Mathematical Reviews (MathSciNet): MR92f:58060
[L1] E. Lerman, A convexity theorem for torus actions on contact manifolds.
[L2] —, Contact toric manifolds.
[LeS] E. Lerman and N. Shirokova, Toric integrable geodesic flows.
[M1] R. Ma\~né, Ergodic Theory and Differentiable Dynamics, Ergeb. Math. Grenzgeb. (3) 8, Springer, Berlin, 1987.
Mathematical Reviews (MathSciNet): MR88c:58040
[M2] —, ``On a theorem of Klingenberg'' in Dynamical Systems and Bifurcation Theory (Rio de Janeiro, 1985), Pitman Res. Notes Math. Ser. 160, Longman Sci. Tech., Harlow, England, 1987, 319--345.
Mathematical Reviews (MathSciNet): MR88k:58129
[SZ] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth, to appear in Duke Math. J.
Mathematical Reviews (MathSciNet): MR1924569
Digital Object Identifier: doi:10.1215/S0012-7094-02-11431-8
Project Euclid: euclid.dmj/1087575454
[T1] J. A. Toth, Various quantum mechanical aspects of quadratic forms, J. Funct. Anal. 130 (1995), 1--42.
Mathematical Reviews (MathSciNet): MR96k:58231
Digital Object Identifier: doi:10.1006/jfan.1995.1062
[T2] —, Eigenfunction localization in the quantized rigid body, J. Differential Geom. 43 (1996), 844--858.
Mathematical Reviews (MathSciNet): MR98a:58165
[T3] —, On the quantum expected values of integrable metric forms, J. Differential Geom. 52 (1999), 327--374.
Mathematical Reviews (MathSciNet): MR2001j:58048
[TZ] J. A. Toth and S. Zelditch, \(L^p\)-estimates of eigenfunctions in the completely integrable case, preprint, 2000.
[V] J. M. VanderKam, \(L\sp \infty\) norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices 1997, 329--347., ; Correction, Internat. Math. Res. Notices 1998, 65.
Mathematical Reviews (MathSciNet): MR99d:58175
Mathematical Reviews (MathSciNet): MR99d:58176
Digital Object Identifier: doi:10.1155/S1073792897000238
[W] J. A. Wolf, Spaces of Constant Curvature, 5th ed., Publish or Perish, Houston, 1984.
Mathematical Reviews (MathSciNet): MR88k:53002
[Y] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc. 22 (1969), 509--512.
Mathematical Reviews (MathSciNet): MR39:7222

2008 © Duke University Press