Hostname: page-component-7479d7b7d-t6hkb Total loading time: 0 Render date: 2024-07-09T03:28:38.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Bound states of a converging quantum waveguide

Published online by Cambridge University Press:  23 November 2012

Giuseppe Cardone ,
Sergei A. Nazarov  and
Keijo Ruotsalainen
Giuseppe Cardone
Affiliation:
University of Sannio - Department of Engineering, Piazza Roma, 21, 84100 Benevento, Italy. giuseppe.cardone@unisannio.it
Sergei A. Nazarov
Affiliation:
Institute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178 St. Petersburg, Russia; srgnazarov@yahoo.co.uk
Keijo Ruotsalainen
Affiliation:
University of Oulu - Department of Electrical Engineering, P.O. Box 4500, 90014 Oulu, Finland; keijo.ruotsalainen@ee.oulu.fi
Get access

Abstract

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+.

Keywords

Quantum waveguidespectrumasymptotics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 305 - 315
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Avishai, Y., Bessis, D., Giraud, B.G. and Mantica, G., Quantum bound states in open geometries. Phys. Rev. B 44 (1991) 8028–8034. Google Scholar
M.Sh. Birman and M.Z. Solomjak, Spectral theory of selfadjoint operators in Hilbert space. Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. Math. Appl. (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987).
Borisov, D., Bunoiu, R. and Cardone, G., On a waveguide with frequently alternating boundary conditions : homogenized Neumann condition. Ann. Henri Poincaré 11 (2010) 1591–1627. Google Scholar
Borisov, D., Bunoiu, R. and Cardone, G., On a waveguide with an infinite number of small windows. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 53–56. Google Scholar
Borisov, D., Bunoiu, R. and Cardone, G., Homogenization and asymptotics for a waveguide with an infinite number of closely located small windows. Prob. Math. Anal. 58 (2011) 59–68; J. Math. Sci. 176 (2011) 774-785. Google Scholar
D. Borisov, R. Bunoiu and G. Cardone, Waveguide with non-periodically alternating Dirichlet and Robin conditions : homogenization and asymptotics. Z. Angew. Math. Phys. (ZAMP), DOI 10.1007/s00033-012-0264-2.
Borisov, D. and Cardone, G., Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A, Math. Theor. 42 (2009) 365205. Google Scholar
Borisov, D. and Cardone, G., Planar Waveguide with “Twisted” Boundary Conditions : Discrete Spectrum. J. Math. Phys. 52 (2011) 123513. Google Scholar
Borisov, D. and Cardone, G., Planar Waveguide with “Twisted” Boundary Conditions : Small Width. J. Math. Phys. 53 (2012) 023503. Google Scholar
Borisov, D., Exner, P., Gadyl’shin, R., and Krejčiřík, D., Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2 (2001) 553–572. Google Scholar
Bulla, W., Gesztesy, F., Renger, W. and Simon, B., Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125 (1997) 1487–1495. Google Scholar
Cardone, G., Minutolo, V. and Nazarov, S.A., Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729–741. Google Scholar
Cardone, G., Nazarov, S.A. and Perugia, C., A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222–1244. Google Scholar
Cardone, G., Nazarov, S.A. and Ruotsalainen, K., Asymptotics of an eigenvalue in the continuous spectrum of a converging waveguide. Mat. Sb. 203 (2012) 3–32. Google Scholar
Cardone, G., Minutolo, V. and Nazarov, S.A., Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729–741. Google Scholar
Cardone, G., Nazarov, S.A. and Perugia, C., A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222–1244. Google Scholar
Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73–102. Google Scholar
Exner, P. and Vugalter, S.A., Bound states in a locally deformed waveguide : the critical case. Lett. Math. Phys. 39 (1997) 59–68. Google Scholar
Gadyl’shin, R.R., On local perturbations of quantum waveguides. (Russian) Teoret. Mat. Fiz. 145 (2005) 358–371; Engl. transl. : Theoret. Math. Phys. 145 (2005) 1678–1690. Google Scholar
Grushin, V.V., On the eigenvalues of a finitely perturbed Laplace operator in infinite cylindrical domains. Mat. Zametki 75 (2004) 360–371; Engl. transl. : Math. Notes 75 (2004) 331–340. Google Scholar
Jones, D.S., The eigenvalues of ∇2u + λu = 0 when the boundary conditions are given on semi-infinite domains. Proc. Cambridge Philos. Soc. 49 (1953) 668–684. Google Scholar
Kondratiev, V.A., Boundary value problems for elliptic problems in domains with conical or corner points, Trudy Moskov. Matem. Obshch 16 (1967) 209–292. Engl. transl. : Trans. Moscow Math. Soc. 16 (1967) 227–313. Google Scholar
Maz’ya, V.G. and Plamenevskii, B.A., On coefficients in asymptotics of solutions of elliptic boundary value problems in a domain with conical points, Math. Nachr. 76 (1977) 29–60; Engl. transl. : Amer. Math. Soc. Transl. 123 (1984) 57–89. Google Scholar
Maz’ya, V.G. and Plamenevskii, B.A., Estimates in L p and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr. 81 (1978) 25–82; Engl. transl. : Amer. Math. Soc. Transl. Ser. 123 (1984) 1–56. Google Scholar
V.G. Maz’ya, S.A. Nazarov and B.A. Plamenevskij, Boris Asymptotic theory of elliptic boundary value problems in singularly perturbed domains II, Translated from the German by Plamenevskij. Operator Theory : Advances and Applications. Birkhäuser Verlag, Basel 112 (2000).
Nazarov, S.A., Two-term asymptotics of solutions of spectral problems with singular perturbations, Mat. sbornik. 178 (1991) 291–320; Engl. transl. : Math. USSR. Sbornik. 69 (1991) 307–340. Google Scholar
Nazarov, S.A., Discrete spectrum of cranked, branchy and periodic waveguides, Algebra i analiz 23 (2011) 206–247; Engl. transl. : St. Petersburg Math. J. 23 (2011). Google Scholar
S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries. Nauka, Moscow (1991); Engl. transl. : Elliptic problems in domains with piecewise smooth boundaries. Walter de Gruyter, Berlin, New York (1994). CrossRef