Skip to main content Accesibility Help
×
×
Home

Bound states of a converging quantum waveguide

  • Giuseppe Cardone (a1), Sergei A. Nazarov (a2) and Keijo Ruotsalainen (a3)
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+.

Copyright
References
Hide All
[1] Avishai, Y., Bessis, D., Giraud, B.G. and Mantica, G., Quantum bound states in open geometries. Phys. Rev. B 44 (1991) 80288034.
[2] 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).
[3] Borisov, D., Bunoiu, R. and Cardone, G., On a waveguide with frequently alternating boundary conditions : homogenized Neumann condition. Ann. Henri Poincaré 11 (2010) 15911627.
[4] 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) 5356.
[5] 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) 5968; J. Math. Sci. 176 (2011) 774-785.
[6] 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.
[7] Borisov, D. and Cardone, G., Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A, Math. Theor. 42 (2009) 365205.
[8] Borisov, D. and Cardone, G., Planar Waveguide with “Twisted” Boundary Conditions : Discrete Spectrum. J. Math. Phys. 52 (2011) 123513.
[9] Borisov, D. and Cardone, G., Planar Waveguide with “Twisted” Boundary Conditions : Small Width. J. Math. Phys. 53 (2012) 023503.
[10] Borisov, D., Exner, P., Gadyl’shin, R., and Krejčiřík, D., Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2 (2001) 553572.
[11] Bulla, W., Gesztesy, F., Renger, W. and Simon, B., Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125 (1997) 14871495.
[12] Cardone, G., Minutolo, V. and Nazarov, S.A., Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729741.
[13] 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) 12221244.
[14] Cardone, G., Nazarov, S.A. and Ruotsalainen, K., Asymptotics of an eigenvalue in the continuous spectrum of a converging waveguide. Mat. Sb. 203 (2012) 332.
[15] Cardone, G., Minutolo, V. and Nazarov, S.A., Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729741.
[16] 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) 12221244.
[17] Duclos, P. and Exner, P., Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73102.
[18] Exner, P. and Vugalter, S.A., Bound states in a locally deformed waveguide : the critical case. Lett. Math. Phys. 39 (1997) 5968.
[19] Gadyl’shin, R.R., On local perturbations of quantum waveguides. (Russian) Teoret. Mat. Fiz. 145 (2005) 358371; Engl. transl. : Theoret. Math. Phys. 145 (2005) 1678–1690.
[20] Grushin, V.V., On the eigenvalues of a finitely perturbed Laplace operator in infinite cylindrical domains. Mat. Zametki 75 (2004) 360371; Engl. transl. : Math. Notes 75 (2004) 331–340.
[21] 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) 668684.
[22] Kondratiev, V.A., Boundary value problems for elliptic problems in domains with conical or corner points, Trudy Moskov. Matem. Obshch 16 (1967) 209292. Engl. transl. : Trans. Moscow Math. Soc. 16 (1967) 227–313.
[23] 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) 2960; Engl. transl. : Amer. Math. Soc. Transl. 123 (1984) 57–89.
[24] 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) 2582; Engl. transl. : Amer. Math. Soc. Transl. Ser. 123 (1984) 1–56.
[25] 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).
[26] Nazarov, S.A., Two-term asymptotics of solutions of spectral problems with singular perturbations, Mat. sbornik. 178 (1991) 291320; Engl. transl. : Math. USSR. Sbornik. 69 (1991) 307–340.
[27] Nazarov, S.A., Discrete spectrum of cranked, branchy and periodic waveguides, Algebra i analiz 23 (2011) 206247; Engl. transl. : St. Petersburg Math. J. 23 (2011).
[28] 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).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

ESAIM: Mathematical Modelling and Numerical Analysis
  • ISSN: 0764-583X
  • EISSN: 1290-3841
  • URL: /core/journals/esaim-mathematical-modelling-and-numerical-analysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed