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 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 }, pages = {305--315}, publisher = {EDP-Sciences}, volume = {47}, number = {1}, year = {2013}, doi = {10.1051/m2an/2012033}, mrnumber = {2997503}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012033/} }
TY - JOUR AU - Cardone, Giuseppe AU - Nazarov, Sergei A. AU - Ruotsalainen, Keijo TI - Bound states of a converging quantum waveguide JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 305 EP - 315 VL - 47 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012033/ DO - 10.1051/m2an/2012033 LA - en ID - M2AN_2013__47_1_305_0 ER -
%0 Journal Article %A Cardone, Giuseppe %A Nazarov, Sergei A. %A Ruotsalainen, Keijo %T Bound states of a converging quantum waveguide %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 305-315 %V 47 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012033/ %R 10.1051/m2an/2012033 %G en %F 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 , Volume 47 (2013) no. 1, pp. 305-315. doi : 10.1051/m2an/2012033. http://www.numdam.org/articles/10.1051/m2an/2012033/
[1] Quantum bound states in open geometries. Phys. Rev. B 44 (1991) 8028-8034.
, , and ,[2] 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 | Zbl
and ,[3] On a waveguide with frequently alternating boundary conditions : homogenized Neumann condition. Ann. Henri Poincaré 11 (2010) 1591-1627. | MR | Zbl
, and ,[4] On a waveguide with an infinite number of small windows. C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011) 53-56. | MR | Zbl
, and ,[5] 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 | Zbl
, and ,[6] 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 | Zbl
, and ,[7] Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A, Math. Theor. 42 (2009) 365205. | MR | Zbl
and ,[8] Planar Waveguide with “Twisted” Boundary Conditions : Discrete Spectrum. J. Math. Phys. 52 (2011) 123513. | MR | Zbl
and ,[9] Planar Waveguide with “Twisted” Boundary Conditions : Small Width. J. Math. Phys. 53 (2012) 023503. | MR | Zbl
and ,[10] Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2 (2001) 553-572. | MR | Zbl
, , , and ,[11] Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc. 125 (1997) 1487-1495. | MR | Zbl
, , and ,[12] Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729-741. | MR | Zbl
, and ,[13] A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222-1244. | MR | Zbl
, and ,[14] Asymptotics of an eigenvalue in the continuous spectrum of a converging waveguide. Mat. Sb. 203 (2012) 3-32. | MR | Zbl
, and ,[15] Gaps in the essential spectrum of periodic elastic waveguides. Z. Angew. Math. Mech. 89 (2009) 729-741. | MR | Zbl
, and ,[16] A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface. Math. Nach. 283 (2010) 1222-1244. | MR | Zbl
, and ,[17] Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7 (1995) 73-102. | MR | Zbl
and ,[18] Bound states in a locally deformed waveguide : the critical case. Lett. Math. Phys. 39 (1997) 59-68. | MR | Zbl
and ,[19] On local perturbations of quantum waveguides. (Russian) Teoret. Mat. Fiz. 145 (2005) 358-371; Engl. transl. : Theoret. Math. Phys. 145 (2005) 1678-1690. | MR | Zbl
,[20] 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 | Zbl
,[21] The eigenvalues of when the boundary conditions are given on semi-infinite domains. Proc. Cambridge Philos. Soc. 49 (1953) 668-684. | MR | Zbl
,[22] 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
,[23] 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 | Zbl
and ,[24] 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 | Zbl
and ,[25] 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 | Zbl
, and ,[26] 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
,[27] Discrete spectrum of cranked, branchy and periodic waveguides, Algebra i analiz 23 (2011) 206-247; Engl. transl. : St. Petersburg Math. J. 23 (2011). | MR | Zbl
,[28] 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 | Zbl
and ,Cited by Sources: