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Toth, John A.; Zelditch, Steve

Riemannian manifolds with uniformly bounded eigenfunctions. (English) Zbl 1022.58013

Duke Math. J. 111, No. 1, 97-132 (2002).
Let \(V_\lambda= \{\phi: \Delta\phi_\lambda= \lambda\phi_\lambda\}\) denote the \(\lambda\)-eigenspace for \(\lambda\) in the spectrum of the Laplacian on a compact Riemannian manifold \((M,g)\). Define \[ L^\infty(\lambda, g)= \sup_{\substack{ \phi\in V_\lambda\\ \|\phi\|_{L^2}= 1}} \|\phi\|_{L^\infty},\quad l^\infty(\lambda, g)= \inf_{\text{ONB}\{\phi_j\}\in V_\lambda} \Biggl(\sup_{j= 1,\dots,\dim V_\lambda}\|\phi_j\|_{L^\infty}\Biggr), \] where ONB denotes orthonormal basis. The aim of the article is to determine the \((M,g)\) for which \(l^\infty(\lambda, g)={\mathcal O}(1)\) and those for which \(L^\infty(\lambda, g)={\mathcal O}(1)\). It is found that for quantum completely integrable Laplacians under further suitable conditions the manifold \((M,g)\) is flat.
Related questions for semiclassical Schrödinger operators \(\hbar^2\Delta+ V\) are analysed.
Reviewer: Klaus Kirsten (Manchester)

Cited in 1 Review
Cited in 21 Documents

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53D25 Geodesic flows in symplectic geometry and contact geometry

Keywords:

eigenfunctions; bounds; Laplacian; Schrödinger operators
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\textit{J. A. Toth} and \textit{S. Zelditch}, Duke Math. J. 111, No. 1, 97--132 (2002; Zbl 1022.58013)
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References:

[1] R. Abraham and J. E. Marsden, Foundations of Mechanics , 2d ed., Benjamin/Cummings, Reading, Mass., 1978. · Zbl 0393.70001
[2] V. I. Arnold, Modes and quasimodes (in Russian), Funktsional. Anal. i Priložen. 6 , no. 2 (1972), 12–20.; English translation in Funct. Anal. Appl. 6 (1972), 94–101.
[3] M. V. Berry, Regular and irregular semiclassical wavefunctions , J. Phys. A 10 (1977), 2083–2091. · Zbl 0377.70014
[4] –. –. –. –., Semi-classical mechanics in phase space: A study of Wigner’s function , Philos. Trans. Roy. Soc. London Ser. A 287 (1977), 237–271. JSTOR: · Zbl 0421.70020
[5] M. Bialy and L. Polterovich, Hopf-type rigidity for Newton equations , Math. Res. Lett. 2 (1995), 695–700. · Zbl 0861.58012
[6] 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. · Zbl 0842.58072
[7] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat , Geom. Funct. Anal. 4 (1994), 259–269. · Zbl 0808.53038
[8] A.-M. Charbonnel, Comportement semi-classique du spectre conjoint d’opérateurs pseudodifférentiels qui commutent , Asympt. Anal. 1 (1988), 227–261. · Zbl 0665.35080
[9] Y. Colin de Verdière, Quasi-modes sur les variétés riemanniennes , Invent. Math. 43 (1977), 15–52. · Zbl 0449.53040
[10] –. –. –. –., Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques , Comment. Math. Helv. 54 (1979), 508–522. · Zbl 0459.58014
[11] –. –. –. –., Spectre conjoint d’opérateurs pseudo-différentiels qui commutent, II: Le cas intégrable , Math. Z. 171 (1980), 51–73. · Zbl 0478.35073
[12] Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique, I: Concentration microlocale , Comm. Partial Differential Equations 19 (1994), 1535–1563. · Zbl 0819.35116
[13] C. Croke and B. Kleiner, On tori without conjugate points , Invent. Math. 120 (1995), 241–257. · Zbl 0842.53028
[14] J. J. Duistermaat, On global action-angle coordinates , Comm. Pure Appl. Math. 33 (1980), 687–706. · Zbl 0439.58014
[15] A. Einstein, Zum Quantensatz von Sommerfeld und Epstein , Verh. Deutsch. Phys. Ges. 19 (1917), 82–92.
[16] V. Guillemin and S. Sternberg, Geometric Asymptotics , Math. Surveys 14 , Amer. Math. Soc., Providence, 1977. · Zbl 0364.53011
[17] –. –. –. –., Homogeneous quantization and multiplicities of group representations , J. Funct. Anal. 47 (1982), 344–380. · Zbl 0733.58021
[18] G. J. Heckman, Quantum integrability for the Kovalevsky top , Indag. Math. (N.S.) 9 (1998), 359–365. · Zbl 0939.37037
[19] B. Helffer, Semi-classical Analysis for the Schrödinger Operator and Applications , Lecture Notes in Math. 1336 , Springer, Berlin, 1988. · Zbl 0647.35002
[20] L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators , Grundlehren. Math. Wiss. 275 , Springer, Berlin, 1985. · Zbl 0612.35001
[21] 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. · Zbl 1071.58505
[22] A. Knauf, Closed orbits and converse KAM theory , Nonlinearity 3 (1990), 961–973. · Zbl 0702.70013
[23] 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. · Zbl 0838.58041
[24] F. Lalonde and J.-C. Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactes des fibrés cotangents , Comment. Math. Helv. 66 (1991), 18–33. · Zbl 0759.53022
[25] E. Lerman, A convexity theorem for torus actions on contact manifolds , · Zbl 1021.53061
[26] ——–, Contact toric manifolds , · Zbl 1079.53118
[27] E. Lerman and N. Shirokova, Toric integrable geodesic flows , · Zbl 1001.37046
[28] R. Mañé, Ergodic Theory and Differentiable Dynamics , Ergeb. Math. Grenzgeb. (3) 8 , Springer, Berlin, 1987.
[29] –. –. –. –., “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. · Zbl 0633.58022
[30] C. D. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunction growth , to appear in Duke Math. J. · Zbl 1018.58010
[31] J. A. Toth, Various quantum mechanical aspects of quadratic forms , J. Funct. Anal. 130 (1995), 1–42. · Zbl 0841.58029
[32] –. –. –. –., Eigenfunction localization in the quantized rigid body , J. Differential Geom. 43 (1996), 844–858. · Zbl 0871.58050
[33] –. –. –. –., On the quantum expected values of integrable metric forms , J. Differential Geom. 52 (1999), 327–374. · Zbl 0992.53063
[34] J. A. Toth and S. Zelditch, (L^p )-estimates of eigenfunctions in the completely integrable case, preprint, 2000. · Zbl 1028.58028
[35] 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): · Zbl 0877.58056
[36] J. A. Wolf, Spaces of Constant Curvature , 5th ed., Publish or Perish, Houston, 1984. · Zbl 0556.53033
[37] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant , Proc. Amer. Math. Soc. 22 (1969), 509–512. · Zbl 0184.12103
[38] S. Zelditch, Quantum transition amplitudes for ergodic and for completely integrable systems , J. Funct. Anal. 94 (1990), 415–436. · Zbl 0721.58051
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