Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 108, 12 pages      arXiv:0711.3347
Contribution to the Proceedings of the 3-rd Microconference Analytic and Algebraic Methods III

Straight Quantum Waveguide with Robin Boundary Conditions

Martin Jílek
Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Brehová 7, 11519 Prague, Czech Republic

Received August 10, 2007, in final form November 08, 2007; Published online November 21, 2007

We investigate spectral properties of a quantum particle confined to an infinite straight planar strip by imposing Robin boundary conditions with variable coupling. Assuming that the coupling function tends to a constant at infinity, we localize the essential spectrum and derive a sufficient condition which guarantees the existence of bound states. Further properties of the associated eigenvalues and eigenfunctions are studied numerically by the mode-matching technique.

Key words: quantum waveguides; bound states; Robin boundary conditions.

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  1. Adams R.A., Sobolev spaces, Academic Press, New York, 1975.
  2. Bendali A., Lemrabet K., The effect of a thin coating on the scattering of a time-harmonic wave for the Helmholtz equation, SIAM J. Appl. Math. 56 (1996), 1664-1693.
  3. Borisov D., Exner P., Gadyl'shin R., Krejcirík D., Bound states in weakly deformed strips and layers, Ann. Henri Poincaré 2 (2001), 553-572, math-ph/0011052.
  4. Borisov D., Krejcirík D., PT-symmetric waveguide, arXiv:0707.3039.
  5. Bulla W., Gesztesy F., Renger W., Simon B., Weakly coupled boundstates in quantum waveguides, Proc. Amer. Math. Soc. 127 (1997), 1487-1495.
  6. Chenaud B., Duclos P., Freitas P., Krejcirík D., Geometrically induced discrete spectrum in curved tubes, Differential Geom. Appl. 23 (2005), 95-105, math.SP/0412132.
  7. Davies E.B., Spectral theory and differential operators, Camb. Univ. Press, Cambridge, 1995.
  8. Dermenjian Y., Durand M., Iftimie V., Spectral analysis of an acoustic multistratified perturbed cylinder, Comm. Partial Differential Equations 23 (1998), 141-169.
  9. Dittrich J., Kríz J., Bound states in straight quantum waveguides with combined boundary conditions, J. Math. Phys. 43 (2002) 3892-3915, math-ph/0112018.
  10. Dittrich J., Kríz J., Curved planar quantum wires with Dirichlet and Neumann boundary conditions, J. Phys. A: Math. Gen. 35 (2002), L269-L275, math-ph/0203007.
  11. Duclos P., Exner P., Curvature-induced bound states in quantum waveguides in two and three dimensions, Rev. Math. Phys. 7 (1995), 73-102.
  12. Engquist B., Nedelec J.C., Effective boundary conditions for electro-magnetic scattering in thin layers, Rapport Interne, Vol. 278, CMAP, 1993.
  13. Evans L.C., Partial differential equations, American Mathematical Society, Providence, 1998.
  14. Exner P., Krejcirík D., Quantum waveguides with a lateral semitransparent barrier: spectral and scattering properties, J. Phys. A: Math. Gen. 32 (1999), 4475-4494, cond-mat/9904379.
  15. Exner P., Seba P., Bound states in curved quantum waveguides, J. Math. Phys. 30 (1989), 2574-2580.
  16. Exner P., Seba P., Tater M., Vanek D., Bound states and scattering in quantum waveguides coupled laterally through a boundary window, J. Math. Phys. 37 (1996), 4867-4887, cond-mat/9512088.
  17. Freitas P., Krejcirík D., Waveguides with combined Dirichlet and Robin boundary conditions, Math. Phys. Anal. Geom. 9 (2006), 335-352, math-ph/0701075.
  18. Goldstone J., Jaffe R.L., Bound states in twisting tubes, Phys. Rev. B 45 (1992), 14100-14107.
  19. Kato T., Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966.
  20. Kovarík H., Krejcirík D., A Hardy inequality in a twisted Dirichlet-Neumann waveguide, Math. Nachr., to appear, math-ph/0603076.
  21. Krejcirík D., Kríz J., On the spectrum of curved planar waveguides, Publ. RIMS Kyoto Univ. 41 (2005), 757-791.
  22. Lin K., Jaffe R.L., Bound states and quantum resonances in quantum wires with circular bends, Phys. Rev. B 54 (1996), 5750-5762.
  23. Londergan J.T., Carini J.P., Murdock D.P., Binding and scattering in two-dimensional systems, Lect. Note in Phys., Vol. 60, Springer, Berlin, 1999.
  24. Olendski O., Mikhailovska L., Localized-mode evolution in a curved planar waveguide with combined Dirichlet and Neumann boundary conditions, Phys. Rev. E 67 (2003), 056625, 11 pages.
  25. Reed M., Simon B., Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, New York, 1978.

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