Bound states of a converging quantum waveguide
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 305-315
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+.
DOI : https://doi.org/10.1051/m2an/2012033
Classification:  35P15,  47A75,  49R50
Keywords: quantum waveguide, spectrum, asymptotics
@article{M2AN_2013__47_1_305_0,
     author = {Cardone, Giuseppe and Nazarov, Sergei A. and Ruotsalainen, Keijo},
     title = {Bound states of a converging quantum waveguide},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {305-315},
     doi = {10.1051/m2an/2012033},
     mrnumber = {2997503},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_1_305_0}
}
Cardone, Giuseppe; Nazarov, Sergei A.; Ruotsalainen, Keijo. Bound states of a converging quantum waveguide. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 305-315. doi : 10.1051/m2an/2012033. http://www.numdam.org/item/M2AN_2013__47_1_305_0/

[1] Y. Avishai, D. Bessis, B.G. Giraud and G. Mantica, Quantum bound states in open geometries. Phys. Rev. B 44 (1991) 8028-8034.

[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). | MR 609148 | Zbl 0744.47017

[3] D. Borisov, R. Bunoiu and G. Cardone, On a waveguide with frequently alternating boundary conditions : homogenized Neumann condition. Ann. Henri Poincaré 11 (2010) 1591-1627. | MR 2769705 | Zbl 1210.82077

[4] D. Borisov, R. Bunoiu and G. Cardone, On a waveguide with an infinite number of small windows. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 53-56. | MR 2755696 | Zbl 1211.35098

[5] D. Borisov, R. Bunoiu and G. Cardone, 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. | MR 2838974 | Zbl 1290.81038

[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. | MR 3068832 | Zbl 1282.35033

[7] D. Borisov and G. Cardone, Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A, Math. Theor. 42 (2009) 365205. | MR 2534513 | Zbl 1178.81088

[8] D. Borisov and G. Cardone, Planar Waveguide with “Twisted” Boundary Conditions : Discrete Spectrum. J. Math. Phys. 52 (2011) 123513. | MR 2907657 | Zbl 1273.81100

[9] D. Borisov and G. Cardone, Planar Waveguide with “Twisted” Boundary Conditions : Small Width. J. Math. Phys. 53 (2012) 023503. | MR 2920471 | Zbl 1274.81108

[10] D. Borisov, P. Exner, R. Gadyl'Shin, and D. Krejčiřík, Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2 (2001) 553-572. | MR 1846856 | Zbl 1043.35046

[11] W. Bulla, F. Gesztesy, W. Renger and B. Simon, Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125 (1997) 1487-1495. | MR 1371117 | Zbl 0868.35080

[12] G. Cardone, V. Minutolo and S.A. Nazarov, Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729-741. | MR 2567306 | Zbl 1189.35201

[13] G. Cardone, S.A. Nazarov and C. Perugia, A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222-1244. | MR 2730490 | Zbl 1213.35327

[14] G. Cardone, S.A. Nazarov and K. Ruotsalainen, Asymptotics of an eigenvalue in the continuous spectrum of a converging waveguide. Mat. Sb. 203 (2012) 3-32. | MR 2962604 | Zbl 1238.35071

[15] G. Cardone, V. Minutolo and S.A. Nazarov, Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729-741. | MR 2567306 | Zbl 1189.35201

[16] G. Cardone, S.A. Nazarov and C. Perugia, A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222-1244. | MR 2730490 | Zbl 1213.35327

[17] P. Duclos and P. Exner, Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73-102. | MR 1310767 | Zbl 0837.35037

[18] P. Exner and S.A. Vugalter, Bound states in a locally deformed waveguide : the critical case. Lett. Math. Phys. 39 (1997) 59-68. | MR 1432793 | Zbl 0871.35067

[19] R.R. Gadyl'Shin, On local perturbations of quantum waveguides. (Russian) Teoret. Mat. Fiz. 145 (2005) 358-371; Engl. transl. : Theoret. Math. Phys. 145 (2005) 1678-1690. | MR 2243437 | Zbl 1178.81078

[20] V.V. Grushin, 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. | MR 2068799 | Zbl 1111.35022

[21] D.S. Jones, The eigenvalues of 2 u + λ u = 0 when the boundary conditions are given on semi-infinite domains. Proc. Cambridge Philos. Soc. 49 (1953) 668-684. | MR 58086 | Zbl 0051.07704

[22] V.A. Kondratiev, 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. | Zbl 0194.13405

[23] V.G. Maz'Ya and B.A. Plamenevskii, 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. | MR 601608 | Zbl 0554.35036

[24] V.G. Maz'Ya and B.A. Plamenevskii, Estimates in Lp 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. | MR 492821 | Zbl 0554.35035

[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). | MR 1779978 | Zbl 1127.35301

[26] S.A. Nazarov, 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. | Zbl 0732.35004

[27] S.A. Nazarov, Discrete spectrum of cranked, branchy and periodic waveguides, Algebra i analiz 23 (2011) 206-247; Engl. transl. : St. Petersburg Math. J. 23 (2011). | MR 2841676 | Zbl 1238.35075

[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). | MR 1283387 | Zbl 0806.35001