Mathematische Annalen

, Volume 367, Issue 3–4, pp 1647–1683 | Cite as

The adiabatic limit of Schrödinger operators on fibre bundles

  • Jonas Lampart
  • Stefan Teufel


We consider Schrödinger operators \(H=-\Delta _{g_\varepsilon } + V\) on a fibre bundle \(M\mathop {\rightarrow }\limits ^{\pi }B\) with compact fibres and a metric \(g_\varepsilon \) that blows up directions perpendicular to the fibres by a factor \({\varepsilon ^{-1}\gg 1}\). We show that for an eigenvalue \(\lambda \) of the fibre-wise part of H, satisfying a local gap condition, and every \(N\in \mathbb {N}\) there exists a subspace of \(L^2(M)\) that is invariant under H up to errors of order \(\varepsilon ^{N+1}\). The dynamical and spectral features of H on this subspace can be described by an effective operator on the fibre-wise \(\lambda \)-eigenspace bundle \(\mathcal {E}\rightarrow B\), giving detailed asymptotics for H.


  1. 1.
    Baider, A.: Noncompact Riemannian manifolds with discrete spectra. J. Differ. Geom. 14(1), 41–58 (1979)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bando, S., Urakawa, H.: Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds. Tohoku Math. J. 35(2), 155–172 (1983)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bessa, G.P., Montenegro, J.F., Piccione, P.: Riemannian submersions with discrete spectrum. J. Geom. Anal. 22(2), 603–620 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bismut, J.-M., Cheeger, J.: \(\eta \)-invariants and their adiabatic limits. J. Am. Math. Soc. 2(1), 33–70 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bordoni, M.: Spectral estimates for submersions with fibers of basic mean curvature. An. Univ. Vest Timiş. Ser. Mat.-Inform. 44(1), 23–36 (2011)Google Scholar
  6. 6.
    Bouchitté, G., Mascarenhas, M.L., Trabucho, L.: On the curvature and torsion effects in one dimensional waveguides. ESAIM Control Optim. Calc. Var. 13(4), 793–808 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in quantum waveguides. Proc. Am. Math. Soc. 125(5), 1487–1495 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carron, G., Exner, P., Krejčiřík, D.: Topologically nontrivial quantum layers. J. Math. Phys. 45(2), 774–784 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am. J. Math. 92(1), 61–74 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chenaud, B., Duclos, P., Freitas, P., Krejčiřík, D.: Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23(2), 95–105 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Da Costa, R.: Constraints in quantum mechanics. Phys. Rev. A 25(6), 2893 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dai, X.: Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence. J. Am. Math. Soc. 4(2), 265–321 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    De Oliveira, C.R.: Quantum singular operator limits of thin Dirichlet tubes via \(\Gamma \)-convergence. Rep. Math. Phys. 67(1), 1–32 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    de Oliveira, C.R., Verri, A.A.: On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes. J. Math. Anal. Appl. 381(1), 454–468 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7(01), 73–102 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Eichhorn, J.: Global Analysis on Open Manifolds. Nova Science Publishers, Hauppauge (2007)Google Scholar
  17. 17.
    Fermanian-Kammerer, C., Gérard, P.: Mesures semi-classiques et croisement de modes. B. Soc. Math. Fr. 130(1), 123–168 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Freitas, P., Krejčiřík, D.: Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57, 343–376 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Friedlander, L., Solomyak, M.: On the spectrum of narrow periodic waveguides. Russ. J. Math. Phys. 15(2), 238–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Froese, R., Herbst, I.: Realizing holonomic constraints in classical and quantum mechanics. Commun. Math. Phys. 220(3), 489–535 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gadyl’shin, R.R.: Local perturbations of quantum waveguides. Theoret. Math. Phys. 145(3), 1678–1690 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goette, S.: Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds. J. Eur. Math. Soc. 16, 2499–2555 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Goldstone, J., Jaffe, R.L.: Bound states in twisting tubes. Phys. Rev. B (3) 45(24), 14100–14107 (1992)Google Scholar
  24. 24.
    Grieser, D.: Thin tubes in mathematical physics, global analysis and spectral geometry. In: Exner, P., et al. (eds.) Analysis on Graphs and its Applications. Proceedings of Symposia in Pure Mathematics. American Mathematical Society, USA (2008)Google Scholar
  25. 25.
    Grieser, D., Jerison, D.: Asymptotics of the first nodal line of a convex domain. Invent. Math. 125(2), 197–219 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Grushin, V.V.: Asymptotic behavior of eigenvalues of the Laplace operator in thin infinite tubes. Math. Notes 85(5–6), 661–673 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Haag, S., Lampart, J., Teufel, S.: Generalised quantum waveguides. Ann. Henri Poincaré 16(11), 2535–2568 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper (avec application a l’étude de Schrödinger avec champ magnétique). Mém. Soc. Math. France 34, 1761–1771 (1988)zbMATHGoogle Scholar
  29. 29.
    Helffer, B., Sjöstrand, J.: Analyse semi-classique pour l’équation de Harper. II.: Comportement semi-classique près d’un rationnel. Mém. Soc. Math. France 40, 1–139 (1990)zbMATHGoogle Scholar
  30. 30.
    Jerison, D.: The diameter of the first nodal line of a convex domain. Ann. Math. (2) 141(1), 1–33 (1995)Google Scholar
  31. 31.
    Jerison, D.: The first nodal set of a convex domain. In: Fefferman, C., Fefferman, R., Wainger, S. (eds.) Essays on Fourier Analysis in Honor of Elias M. Stein, pp. 225–249. Princeton University Press, Princeton (1995)Google Scholar
  32. 32.
    Kleine, R.: Discreteness conditions for the Laplacian on complete, non-compact Riemannian manifolds. Math. Z. 198(1), 127–141 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kolb, M., Krejčiřík, D.: The Brownian traveller on manifolds. J. Spectr. Theory 4, 235–281 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kordyukov, Y.A.: Adiabatic limits and spectral geometry of foliations. Math. Ann. 313(4), 763–783 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kovařík, H., Vugalter, S.: Estimates on trapped modes in deformed quantum layers. J. Math. Anal. Appl. 345(1), 566–572 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Krejčiřík, D.: Quantum strips on surfaces. J. Geom. Phys. 45(1), 203–217 (2003)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Krejčiřík, D., Lu, Z.: Location of the essential spectrum in curved quantum layers. J. Math. Phys. 55(8), 083520 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Krejčiřík, D., Tušek, M.: Nodal sets of thin curved layers. J. Differ. Equ. 258(2), 281–301 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Lampart, J.: The adiabatic limit of Schrödinger operators on fibre bundles. PhD thesis, Universität Tübingen (2014)Google Scholar
  40. 40.
    Lampart, J.: Convergence of nodal sets in the adiabatic limit. Ann. Global Anal. Geom. 47(2), 147–166 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lin, C., Lu, Z.: On the discrete spectrum of generalized quantum tubes. Comm. Partial Differ. Equ. 31(10), 1529–1546 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Álvarez, J.A., López, Y., Kordyukov, A.: Adiabatic limits and spectral sequences for Riemannian foliations. Geom. Funct. Anal. 10(5), 977–1027 (2000)Google Scholar
  43. 43.
    Lott, J.: Collapsing and the differential form Laplacian: the case of a smooth limit space. Duke Math. J. 114(2), 267–306 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Maraner, P.: A complete perturbative expansion for quantum mechanics with constraints. J. Phys. A 28(10), 2939 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Martinez, A., Sordoni, V.: A general reduction scheme for the time-dependent Born–Oppenheimer approximation. C. R. Math. Acad. Sci. Paris 334(3), 185–188 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Mazzeo, R.R., Melrose, R.B.: The adiabatic limit, Hodge cohomology and Leray’s spectral sequence for a fibration. J. Differ. Geom. 31(1), 185–213 (1990)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Mitchell, K.A.: Gauge fields and extrapotentials in constrained quantum systems. Phys. Rev. A 63(4), 042112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Nenciu, G.: Linear adiabatic theory. Exponential estimates. Commun. Math. Phys. 152(3), 479–496 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Panati, G., Spohn, H., Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242(3), 547–578 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Panti, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7(1), 145–204 (2003)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Post, O.: Spectral Analysis on Graph-like Spaces. Lecture Notes in Mathematics. Springer, Berlin (2012)CrossRefzbMATHGoogle Scholar
  52. 52.
    Reed, M., Simon, B.: Methods of modern mathematical physics: II Fourier analysis, self-adjointness. Academic Press, New York (1975)Google Scholar
  53. 53.
    Schick, T.: Analysis on \(\partial \)-manifolds of bounded geometry. In: Hodge-De Rham Isomorphism and \(L^2\)-Index Theorem. Shaker, OH (1996)Google Scholar
  54. 54.
    Schick, T.: Manifolds with boundary and of bounded geometry. Math. Nachr. 223, 89–102 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology. International Press, USA (1994)Google Scholar
  56. 56.
    Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Lecture Notes in Mathematics. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  57. 57.
    Wachsmuth, J., Teufel, S.: Effective Hamiltonians for constrained quantum systems. Mem. Am. Math. Soc. 230, 2013 (1083)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Wittich, O.: \(L^2\)-homogenization of heat equations on tubular neighborhoods (2008). arXiv:0810.5047

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.PSL Research University and CEREMADE (UMR CNRS 7534)Université de Paris-DauphineParis Cedex 16France
  2. 2.Fachbereich MathematikUniversität TübingenTübingenGermany

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