The Journal of Geometric Analysis

, Volume 25, Issue 4, pp 2546–2564 | Cite as

The Magnetic Laplacian in Shrinking Tubular Neighborhoods of Hypersurfaces



The Dirichlet Laplacian between two parallel hypersurfaces in Euclidean spaces of any dimension in the presence of a magnetic field is considered in the limit when the distance between the hypersurfaces tends to zero. We show that the Laplacian converges in a norm-resolvent sense to a Schrödinger operator on the limiting hypersurface whose electromagnetic potential is expressed in terms of principal curvatures and the projection of the ambient vector potential to the hypersurface. As an application, we obtain an effective approximation of bound-state energies and eigenfunctions in thin quantum layers.


Curvature of hypersurfaces Effective potential Eigenvalue asymptotics 

Mathematics Subject Classification

35J25 35B25 35B40 58J50 81Q10 


  1. 1.
    Carron, G., Exner, P., Krejčiřík, D.: Topologically nontrivial quantum layers. J. Math. Phys. 45, 774–784 (2004)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    da Costa, R.C.T.: Quantum mechanics of a constrained particle. Phys. Rev. A 23, 1982–1987 (1981)MathSciNetCrossRefGoogle Scholar
  3. 3.
    da Costa, R.C.T.: Constraints in quantum mechanics. Phys. Rev. A 25, 2893–2900 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7, 73–102 (1995)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Duclos, P., Exner, P., Krejčiřík, D.: Bound states in curved quantum layers. Commun. Math. Phys. 223, 13–28 (2001)MATHCrossRefGoogle Scholar
  6. 6.
    Duclos, P., S\(\check{\text{ t }}\)ovíček, P., Tušek, M.: On the two-dimensional Coulomb-like potential with a central point interaction. J. Phys. A 43, 474020 (2010)Google Scholar
  7. 7.
    Ekholm, T., Kovařík, H.: Stability of the magnetic Schrödinger operator in a waveguide. Commun. Partial Differ. Equ. 30(4), 539–565 (2005)MATHCrossRefGoogle Scholar
  8. 8.
    Exner, P., Krejčiřík, D.: Bound states in mildly curved layers. J. Phys. A 34, 5969–5985 (2001)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Exner, P., Šeba, P.: Bound states in curved quantum waveguides. J. Math. Phys. 30, 2574–2580 (1989)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Freitas, P., Krejčiřík, D.: Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57(1), 343–376 (2008)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Froese, R., Herbst, I.: Realizing holonomic constraints in classical and quantum mechanics. Commun. Math. Phys. 220, 489–535 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Friedlander, L., Solomyak, M.: On the spectrum of the Dirichlet Laplacian in a narrow strip. Isr. J. Math. 170, 337–354 (2009)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Friedlander, L., Solomyak, M.: On the spectrum of the Dirichlet Laplacian in a narrow infinite strip. Am. Math. Soc. Transl. 225, 103–116 (2008)MathSciNetGoogle Scholar
  14. 14.
    Hurt, N.E.: Mathematical Physics of Quantum Wires and Devices. Kluwer, Dordrecht (2000)MATHCrossRefGoogle Scholar
  15. 15.
    Jensen, H., Koppe, H.: Quantum mechanics with constraints. Ann. Phys. 63, 586–591 (1971)CrossRefGoogle Scholar
  16. 16.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)MATHCrossRefGoogle Scholar
  17. 17.
    Krejčiřík, D.: Spectrum of the Laplacian in a narrow curved strip with combined Dirichlet and Neumann boundary conditions. ESAIM: Control. Optim. Calcul. Var. 15, 555–568 (2009)MATHCrossRefGoogle Scholar
  18. 18.
    Krejčiřík, D., Kříž, J.: On the spectrum of curved quantum waveguides. Publ. RIMS Kyoto Univ. 41(3), 757–791 (2005)MATHCrossRefGoogle Scholar
  19. 19.
    Krejčiřík, D., Šediváková, H.: The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions. Rev. Math. Phys. 24, 1250018 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kühnel, W.: Differential Geometry. AMS, Providence, RI (2006)Google Scholar
  21. 21.
    Lampart, J., Teufel, S., Wachsmuth, J.: Effective Hamiltonians for thin Dirichlet tubes with varying cross-section. Mathematical Results in Quantum Physics, Hradec Králové, 2010, xi+274 pp., pp. 183–189. World Scientific, Singapore (2011)Google Scholar
  22. 22.
    Laptev, A., Weidl, T.: Hardy inequalities for magnetic Dirichlet forms. Oper. Theory Adv. Appl. 108, 299–305 (1999)MathSciNetGoogle Scholar
  23. 23.
    Lieb, E.H., Loss, M.: Analysis. American Mathematical Society, Providence, RI (1997)Google Scholar
  24. 24.
    Lin, C., Lu, Z.: On the discrete spectrum of generalized quantum tubes. Commun. Partial Differ. Equ. 31, 1529–1546 (2006)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Lin, C., Lu, Z.: Existence of bound states for layers built over hypersurfaces in \(\mathbb{R}^{n+1}\). J. Funct. Anal. 244, 1–25 (2007)MATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Lin, C., Lu, Z.: Quantum layers over surfaces ruled outside a compact set. J. Math. Phys. 48, 053522 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Londergan, J.T., Carini, J.P., Murdock, D.P.: Binding and scattering in two-dimensional systems, LNP, vol. m60. Springer, Berlin (1999)Google Scholar
  28. 28.
    Lu, Z., Rowlett, J.: On the discrete spectrum of quantum layers. J. Math. Phys. 53, 073519 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Mitchell, K.A.: Gauge fields and extrapotentials in constrained quantum systems. Phys. Rev. A 63, 042112 (2001)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Nakahara, M.: Geometry, Topology, and Physics. Taylor & Francis Group, London (2003)MATHGoogle Scholar
  31. 31.
    Simon, B.: Quantum Mechanics for Hamiltonians Defined by Quadratic Forms. Princeton University Press, Princeton (1971)Google Scholar
  32. 32.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II. Publish or Perish, Houston (1999)MATHGoogle Scholar
  33. 33.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I. Publish or Perish, Houston (2005)Google Scholar
  34. 34.
    Teschl, G.: Mathematical Methods in Quantum Mechanics. AMS, Providence, RI (2009)MATHCrossRefGoogle Scholar
  35. 35.
    Tolar, J.: On a Quantum Mechanical d’Alembert Principle, Group Theoretical Methods in Physics, LNP, vol. 313. Springer, Berlin (1988)Google Scholar
  36. 36.
    Wachsmuth, J., Teufel, S.: Effective Hamiltonians for constrained quantum systems. Mem. Am. Math. Soc. 1083, (2013)Google Scholar
  37. 37.
    Wachsmuth, J., Teufel, S.: Constrained quantum systems as an adiabatic problem. Phys. Rev. A 82, 022112 (2010)CrossRefGoogle Scholar
  38. 38.
    Wittich, O.: \(L^2\)-homogenization of heat equations on tubular neighborhoods. arXiv:0810.5047 [math.AP] (2008)

Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Department of Theoretical PhysicsNuclear Physics Institute ASCRŘežCzech Republic
  2. 2.Institut de Recherche Mathématique de RennesUniversité de Rennes 1Rennes CedexFrance
  3. 3.Department of Mathematics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePrague 2Czech Republic

Personalised recommendations