|
SIGMA 3 (2007), 108, 12 pages arXiv:0711.3347
https://doi.org/10.3842/SIGMA.2007.108
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
Abstract
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.
pdf (388 kb)
ps (415 kb)
tex (397 kb)
References
- Adams R.A., Sobolev spaces, Academic Press, New York, 1975.
- 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.
- 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.
- Borisov D., Krejcirík D., PT-symmetric waveguide,
arXiv:0707.3039.
- Bulla W., Gesztesy F., Renger W., Simon B., Weakly coupled
boundstates in quantum waveguides, Proc. Amer. Math. Soc.
127 (1997), 1487-1495.
- 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.
- Davies E.B., Spectral theory and differential operators, Camb. Univ.
Press, Cambridge, 1995.
- Dermenjian Y., Durand M., Iftimie V., Spectral analysis of an
acoustic multistratified perturbed cylinder, Comm. Partial
Differential Equations 23 (1998), 141-169.
- Dittrich J., Kríz J., Bound states in straight quantum
waveguides with combined boundary conditions, J. Math. Phys.
43 (2002) 3892-3915, math-ph/0112018.
- 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.
- Duclos P., Exner P., Curvature-induced bound states in quantum
waveguides in two and three dimensions, Rev. Math. Phys.
7 (1995), 73-102.
- Engquist B., Nedelec J.C., Effective boundary conditions for
electro-magnetic scattering in thin layers, Rapport Interne,
Vol. 278, CMAP, 1993.
- Evans L.C., Partial differential equations, American Mathematical
Society, Providence, 1998.
- 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.
- Exner P., Seba P., Bound states in curved quantum waveguides,
J. Math. Phys. 30 (1989), 2574-2580.
- 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.
- Freitas P., Krejcirík D., Waveguides with combined
Dirichlet and Robin boundary conditions, Math. Phys. Anal.
Geom. 9 (2006), 335-352, math-ph/0701075.
- Goldstone J., Jaffe R.L., Bound states in twisting tubes,
Phys. Rev. B 45 (1992), 14100-14107.
- Kato T., Perturbation theory for linear operators, Springer-Verlag,
Berlin, 1966.
- Kovarík H., Krejcirík D., A Hardy inequality in a
twisted Dirichlet-Neumann waveguide, Math. Nachr., to appear,
math-ph/0603076.
- Krejcirík D., Kríz J., On the spectrum of curved planar waveguides, Publ. RIMS Kyoto Univ. 41
(2005), 757-791.
- Lin K., Jaffe R.L., Bound states and quantum
resonances in quantum wires with circular bends, Phys. Rev. B
54 (1996), 5750-5762.
- 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.
- 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.
- Reed M., Simon B., Methods of modern mathematical physics.
IV. Analysis of operators, Academic Press, New York, 1978.
|
|