| Issue |
ESAIM: M2AN
Volume 47, Number 1, January-February 2013
|
|
|---|---|---|
| Page(s) | 305 - 315 | |
| DOI | https://doi.org/10.1051/m2an/2012033 | |
| Published online | 23 November 2012 | |
Bound states of a converging quantum waveguide
1
University of Sannio - Department of Engineering,
Piazza Roma, 21, 84100
Benevento,
Italy
giuseppe.cardone@unisannio.it
2
Institute of Mechanical Engineering Problems,
V.O., Bolshoi pr., 61,
199178
St. Petersburg,
Russia
srgnazarov@yahoo.co.uk
3
University of Oulu - Department of Electrical
Engineering, P.O. Box
4500, 90014
Oulu,
Finland
keijo.ruotsalainen@ee.oulu.fi
Received:
2
September
2011
Revised:
14
May
2012
We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 − ε, where ε > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weakly coupled bound state below π2, the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when ε → 0+.
Mathematics Subject Classification: 35P15 / 47A75 / 49R50
Key words: Quantum waveguide / spectrum / asymptotics
© EDP Sciences, SMAI, 2012
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