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Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds

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Abstract

We study the long time dynamics of the Schrödinger equation on Zoll manifolds. We establish criteria under which the solutions of the Schrödinger equation can or cannot concentrate on a given closed geodesic. As an application, we derive some results on the set of semiclassical measures for eigenfunctions of Schrödinger operators: we prove that adding a potential \({V \in C^{\infty} (\mathbb{S}^{d})}\) to the Laplacian on the sphere results in the existence of geodesics \({\gamma}\) such that the uniform measure supported on \({\gamma}\) cannot be obtained as a weak-\({\star}\) accumulation point of the densities \({(|\psi_{n}|^{2} {vol}_{\mathbb{S}^d})}\) for any sequence of eigenfunctions \({(\psi_n)}\) of \({\Delta_{\mathbb{S}^{d}} - V}\). We also show that the same phenomenon occurs for the free Laplacian on certain Zoll surfaces.

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References

  1. Anantharaman N., Fermanian-Kammerer C., Macià F.: Semiclassical completely integrable systems: long-time dynamics and observability via two-microlocal Wigner measures. Am. J. Math. 137, 577–638 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anantharaman N., Macià F.: Semiclassical measures for the Schrödinger equation on the torus. JEMS 16, 1253–1288 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Azagra D., Macià F.: Concentration of symmetric eigenfunctions. Nonlinear Anal. 73, 683–688 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Besse A.: Manifolds All of Whose Geodesics Are Closed, Ergeb. Math., vol. 93. Springer, New York (1978)

    Book  Google Scholar 

  5. Bialy M.L., Polterovich L.V.: Lagrangian singularities of invariant tori of Hamiltonian systems with two degrees of freedom. Invent. Math. 97, 291–303 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bolte J., Schwaibold T.: Stability of wave packet dynamics under perturbations. Phys. Rev. E73, 026223 (2006)

    ADS  MathSciNet  Google Scholar 

  7. Brooks, S., Le Masson, E., Lindenstrauss, E.: Quantum ergodicity and averaging operators on the sphere. arXiv:1505.03887 (2015)

  8. Chazarain J.: Spectre d’un hamiltonien quantique et mécanique classique. CPDE 6, 595–644 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  9. Colin de Verdière Y.: Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques. Comment. Math. Helv. 54, 508–522 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Combescure M., Robert D.: A phase-space study of the quantum Loschmidt Echo in the semiclassical limit. Ann. H. Poincaré 8, 91–108 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diestel, J., Uhl, J.J.: Vector measures. Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)

  12. Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Series, vol. 268. Cambridge University Press, Cambridge (1999)

  13. Duistermaat J.J.: On global action-angle coordinates. Commun. Pure Appl. Math. 33, 687–706 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Duistermaat J.J., Guillemin V.: The spectrum of elliptic operators and periodic bicharacteristics. Invent. Math. 29, 39–79 (1975)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Einsiedler M., Ward T.: Ergodic Theory with a View Towards Number Theory, Graduate Texts in Mathematics, vol. 259. Springer, London (2011)

    MATH  Google Scholar 

  16. Eswarathasan, S., Rivière, G.: Perturbation of the semiclassical Schrödinger equation on negatively curved surfaces. J. Inst. Math. Jussieu (2015). http://dx.doi.org/10.1017/S1474748015000262

  17. Gérard, P.: Mesures semi-classiques et ondes de Bloch, Sem. EDP (Polytechnique) 1990–1991, Exp. 16 (1991)

  18. Gérard P.: Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var. 3, 213–233 (1998)

    Article  MathSciNet  Google Scholar 

  19. Gorin T., Prosen T., Seligman T.H., Zdinaric M.: Dynamics of Loschmidt echoes and fidelity decay. Phys. Rep. 435, 33–156 (2006)

    Article  ADS  Google Scholar 

  20. Goussev, A., Jalabert, R.A., Pastawski, H.M., Wisniacki, D.: Loschmidt echo. Scholarpedia 7(8), 11687. arXiv:1206.6348 (2012)

  21. Gromoll, D., Grove, K.: On metrics on \({{\mathbb{S}^{2}}}\) all of whose geodesics are closed. Invent. Math. 65, 175–177 (1981/1982)

  22. Guillemin V.: The Radon transform on Zoll surfaces. Adv. Math. 22, 85–119 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  23. Guillemin V.: Some spectral results for the Laplace operator with potential on the n-sphere. Adv. Math. 27, 273–286 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Guillemin V.: Some spectral results on rank one symmetric spaces. Adv. Math. 28, 129–137 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  25. Guillemin V.: Band asymptotics in 2 dimension. Adv. Math. 42, 248–282 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hall, M.A., Hitrik, M., Sjöstrand, J.: Spectra for semiclassical operators with periodic bicharacteristics in dimension two. Int. Math. Res. Notices. 2015 10243–10277 (2015)

  27. Hatcher A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  28. Helffer B., Robert D.: Puits de potentiel généralisés et asymptotique semi-classique. Ann. Inst. H. Poincaré Phys. Théor. 41, 291–331 (1984)

    MathSciNet  MATH  Google Scholar 

  29. Helgason S.: Integral Geometry and Radon Transforms. Springer, New York (2011)

    Book  MATH  Google Scholar 

  30. Hitrik M.: Eigenfrequencies for damped wave equations on Zoll manifolds. Asymptot. Anal. 31, 265–277 (2002)

    MathSciNet  MATH  Google Scholar 

  31. Hitrik M., Sjöstrand J.: Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions I. Ann. H. Poincaré 5, 1–73 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hörmander L.: The Analysis of Linear Partial Differential Operators III. Springer, Berlin (1985)

    MATH  Google Scholar 

  33. Hörmander L.: The Analysis of Linear Partial Differential Operators IV. Springer, Berlin (1985)

    MATH  Google Scholar 

  34. Jacquod P., Petitjean C.: Decoherence, entanglement and irreversibility in quantum dynamical systems with few degrees of freedom. Adv. Phys. 58, 67–196 (2009)

    Article  ADS  Google Scholar 

  35. Jakobson, D., Zelditch, S.: Classical limits of eigenfunctions for some completely integrable systems, Emerging applications of number theory (Minneapolis, MN, 1996), pp. 329–354, IMA Math. Appl., vol. 109. Springer, New York (1999)

  36. Küster, B., Ramacher, P.: Quantum ergodicity and symmetry reduction. arXiv:1410.1096 (2014)

  37. Macià F.: Some remarks on quantum limits on Zoll manifolds. CPDE 33, 1137–1146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. Macià F.: Semiclassical measures and the Schrödinger flow on Riemannian manifolds. Nonlinearity 22, 1003–1020 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  39. Macià, F., Rivière, G.: Semiclassical measures for the perturbed Schrödinger equation on \({{\mathbb{T}^{d}}}\) (2015, in preparation)

  40. Moser, J., Zehnder, E.: Notes on dynamical systems, Courant Lecture Notes in Mathematics, vol. 12. American Mathematical Society, Providence (2005)

  41. Ojeda-Valencia D., Villegas-Blas C.: Limiting Eigenvalue Distributions Theorems in Semiclassical 1203 Analysis, Spectral Analysis of Quantum Hamiltonians: Spectral Days. Springer, Berlin (2010)

    Google Scholar 

  42. Paternáin G.P.: Geodesic Flows, Progress in Mathematics, vol. 180. Birkhäuser Boston Inc., Boston (1999)

    Book  Google Scholar 

  43. Peres A.: Stability of quantum motion in chaotic and regular systems. Phys. Rev. A 30, 1610–1615 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  44. Rivière, G.: Long time dynamics of the perturbed Schrödinger equation on negatively curved surfaces. Ann. H. Poincaré. arXiv:1412.4400 (2014)

  45. Ruggiero, R.O.: Dynamics and global geometry of manifolds without conjugate points. Ensaios Mat., vol. 12, Soc. Bras. Mat. (2007)

  46. Schwartz L.: Théorie des distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

  47. Uribe A.: Some spectral results on rank one symmetric spaces. Adv. Math. 58, 285–299 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  48. Uribe A., Zelditch S.: Spectral statistics on Zoll surfaces. Commun. Math. Phys. 154, 313–346 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  49. VanderKam J.M.: \({L^{\infty}}\) norms and quantum ergodicity on the sphere. IMRN 7, 329–347 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  50. Weber, J.: J-holomorphic curves in cotangent bundles and the heat flow, Ph.D. thesis/Dissertation, TU Berlin (1999)

  51. Weinstein A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J. 44, 883–892 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  52. Zelditch S.: Quantum ergodicity on the sphere. Commun. Math. Phys. 146, 61–71 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. Zelditch S.: Maximally degenerate Laplacians. Ann. Inst. Fourier 46, 547–587 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. Zelditch S.: Fine structure of Zoll spectra. J. Funct. Anal. 143, 415–460 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  55. Zelditch, S.: Gaussian beams on Zollmanifolds andmaximally degenerate Laplacians. In: Spectral theory and partial differential equations. ContemporaryMathematics, vol. 640. American Mathematical Society, Providence, RI (2015)

  56. Zworski M.: Semiclassical Analysis, Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012)

    Google Scholar 

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Authors and Affiliations

  1. Universidad Politécnica de Madrid. DCAIN, ETSI Navales, Avda. Arco de la Victoria s/n, 28040, Madrid, Spain

    Fabricio Macià

  2. Laboratoire Paul Painlevé (U.M.R. CNRS 8524), U.F.R. de Mathématiques, Université Lille 1, 59655, Villeneuve d’Ascq Cedex, France

    Gabriel Rivière

Authors
  1. Fabricio Macià
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  2. Gabriel Rivière
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Correspondence to Fabricio Macià.

Additional information

Communicated by S. Zelditch

FM takes part in the visiting faculty program of ICMAT and is partially supported by grants ERC Starting Grant 277778 and MTM2013-41780-P (MEC).

GR is partially supported by the Agence Nationale de la Recherche through the Labex CEMPI (ANR-11-LABX-0007-01) and the ANR project GeRaSic (ANR-13-BS01- 0007-01).

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Macià, F., Rivière, G. Concentration and Non-Concentration for the Schrödinger Evolution on Zoll Manifolds. Commun. Math. Phys. 345, 1019–1054 (2016). https://doi.org/10.1007/s00220-015-2504-8

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