Annales Henri Poincaré

, Volume 2, Issue 3, pp 553–572

Bound States in Weakly Deformed Strips and Layers

Authors

  • D. Borisov
    • Bashkir State Pedagogical University, October Revolution St. 3a, RU-450000 Ufa, Russia, e-mail: borisovDI@ic.bashedu.ru
  • P. Exner
    • Department of Theoretical Physics, Nuclear Physics Institue, Academy of Sciences, CZ-25068 Řež, Czech Republic, e-mail: exner@ujf.cas.cz
  • R. Gadyl'shin
    • Bashkir State Pedagogical University, October Revolution St. 3a, RU-450000 Ufa, Russia, e-mail: gadylshin@bspu.ru
  • D. Krejčiřík
    • Department of Theoretical Physics, Nuclear Physics Institue, Academy of Sciences, CZ-25068 Řež, Czech Republic, e-mail: krejcirik@ujf.cas.cz

DOI: 10.1007/PL00001045

Cite this article as:
Borisov, D., Exner, P., Gadyl'shin, R. et al. Ann. Henri Poincaré (2001) 2: 553. doi:10.1007/PL00001045

Abstract.

We consider Dirichlet Laplacians on straight strips in \( {\Bbb R}^2 \) or layers in \( {\Bbb R}^3 \) with a weak local deformation. First we generalize a result of Bulla et al. to the three-dimensional situation showing that weakly coupled bound states exist if the volume change induced by the deformation is positive;we also derive the leading order of the weak-coupling asymptotics. With the knowledge of the eigenvalue analytic properties, we demonstrate then an alternative method which makes it possible to evaluate the next term in the asymptotic expansion for both the strips and layers. It gives,in particular, a criterion for the bound-state existence in the critical case when the added volume is zero.

Copyright information

© Birkhäuser Verlag Basel, 2001