Abstract
The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed.
Article PDF
Similar content being viewed by others
References
Bonnaillie-Noël V., Dauge M., Popoff N., Raymond N.: Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions. Z. Ange. Math. Phys. 63(2), 203–231 (2012)
Borisov D., Cardone G.: Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods. ESAIM Control Optim. Calc. Var. 17, 887–908 (2011)
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) (electronic)
Carron G., Exner P., Krejčiřík D.: Topologically nontrivial quantum layers. J. Math. Phys. 45(2), 774–784 (2004)
Chenaud B., Duclos P., Freitas P., Krejčiřík D.: Geometrically induced discrete spectrum in curved tubes. Differ. Geom. Appl. 23(2), 95–105 (2005)
Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, study edition. Texts and Monographs in Physics. Springer, Berlin (1987)
de Oliveira C.R.: Quantum singular operator limits of thin Dirichlet tubes via Γ-convergence. Rep. Math. Phys. 66, 375–406 (2010)
Dombrowski, N., Raymond, N.: Semiclassical analysis with vanishing magnetic fields. J. Spectr. Theory 3(3) (2013)
Duclos P., Exner P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7(1), 73–102 (1995)
Duclos P., Exner P., Krejčiřík D.: Bound states in curved quantum layers. Comm. Math. Phys. 223(1), 13–28 (2001)
Ekholm T., Kovařík H.: Stability of the magnetic Schrödinger operator in a waveguide. Comm. Partial Differ. Equ. 30(4–6), 539–565 (2005)
Ekholm T., Kovařík H., Krejčiřík D.: A Hardy inequality in twisted waveguides. Arch. Ration. Mech. Anal. 188(2), 245–264 (2008)
Fournais, S., Helffer, B.: Spectral methods in surface superconductivity. Progress in Nonlinear Differential Equations and their Applications, vol. 77. Birkhäuser Boston Inc., Boston (2010)
Freitas P., Krejčiřík D.: Location of the nodal set for thin curved tubes. Indiana Univ. Math. J. 57(1), 343–375 (2008)
Friedlander L., Solomyak M.: On the spectrum of the Dirichlet Laplacian in a narrow strip. Israeli Math. J. 170(1), 337–354 (2009)
Grushin V.V.: Asymptotic behavior of the eigenvalues of the Schrödinger operator in thin closed tubes. Math. Notes 83, 463–477 (2008)
Grushin V.V.: Asymptotic behavior of the eigenvalues of the Schrödinger operator in thin infinite tubes. Math. Notes 85, 661–673 (2009)
Krejčiřík, D.: Twisting versus bending in quantum waveguides. In: Analysis on Graphs and its Applications, Proc. Sympos. Pure Math., vol. 77, pp. 617–637. Amer. Math. Soc., Providence, RI (2008)
Krejčiřík D., Kříž J.: On the spectrum of curved quantum waveguides. Publ. RIMS Kyoto Univ. 41, 757–791 (2005)
Krejčiřík, D., Šediváková, H.: The effective Hamiltonian in curved quantum waveguides under mild regularity assumptions. Rev. Math. Phys. 24(7) (2012)
Krejčiřík D., Zuazua E.: The Hardy inequality and the heat equation in twisted tubes. J. Math. Pures Appl. 94, 277–303 (2010)
Lampart, J., Teufel, S., Wachsmuth, J.: Effective Hamiltonians for thin Dirichlet tubes with varying cross-section. In: Mathematical Results in Quantum Physics, pp. 183–189. World Sci. Publ., Hackensack (2011)
Lin C., Lu Z.: On the discrete spectrum of generalized quantum tubes. Comm. Partial Differ. Equ. 31(10–12), 1529–1546 (2006)
Lin C., Lu Z.: Existence of bound states for layers built over hypersurfaces in \({\mathbb{R}^{n+1}}\). J. Funct. Anal. 244(1), 1–25 (2007)
Lin, C., Lu, Z.: Quantum layers over surfaces ruled outside a compact set. J. Math. Phys. 48(5), 053522 (2007) (14)
Rowlett, J., Lu, Z.: On the discrete spectrum of quantum layers. J. Math. Phys. 53 (2012)
Wachsmuth, J., Teufel, S.: Effective Hamiltonians for constrained quantum systems. Mem. AMS (2013) (to appear)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nader Masmoudi.
Rights and permissions
About this article
Cite this article
Krejčiřík, D., Raymond, N. Magnetic Effects in Curved Quantum Waveguides. Ann. Henri Poincaré 15, 1993–2024 (2014). https://doi.org/10.1007/s00023-013-0298-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0298-9