Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T19:36:46.206Z Has data issue: false hasContentIssue false
  • English
  • Français

BOUND STATES IN WEAKLY DEFORMED WAVEGUIDES: NUMERICAL VERSUS ANALYTICAL RESULTS

Part of: General mathematical topics and methods in quantum theory Partial differential equations, boundary value problems Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  03 October 2017

PAOLO AMORE Open the ORCID record for PAOLO AMORE [Opens in a new window] ,
JOHN P. BOYD Open the ORCID record for JOHN P. BOYD [Opens in a new window] ,
FRANCISCO M. FERNÁNDEZ Open the ORCID record for FRANCISCO M. FERNÁNDEZ [Opens in a new window] ,
MARTIN JACOBO  and
PETR ZHEVANDROV Open the ORCID record for PETR ZHEVANDROV [Opens in a new window]
PAOLO AMORE*
Affiliation:
Facultad de Ciencias, CUICBAS, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, Mexico email [email protected]
JOHN P. BOYD
Affiliation:
Department of Atmospheric, Oceanic & Space Science, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109, USA email [email protected]
FRANCISCO M. FERNÁNDEZ
Affiliation:
INIFTA (UNLP, CCT La Plata-CONICET), Division Quimica Teorica, Blvd. 113 S/N, Sucursal 4, Casilla de Correo 16, 1900 La Plata, Argentina email [email protected]
MARTIN JACOBO
Affiliation:
Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 340, Colima, Colima, Mexico
PETR ZHEVANDROV
Affiliation:
Facultad de Ciencias Físico-Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, 58030 Morelia, Michoacán, Mexico email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study bound states in weakly deformed and heterogeneous waveguides, and compare analytical predictions using a recently developed perturbative method with precise numerical results for three different configurations: a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken strip. We have found excellent agreement between the analytical and numerical results in all the examples; this provides a numerical verification of the analytical approach.

Keywords

trapped modesbound statesdeformed waveguide

MSC classification

Primary: 81Q15: Perturbation theories for operators and differential equations
Secondary: 35P15: Estimation of eigenvalues, upper and lower bounds 65N25: Eigenvalue problems
Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 200 - 214
Copyright
© 2017 Australian Mathematical Society 

References

Amore, P., “Weakly bound states in heterogeneous waveguides: a calculation to fourth order”. Preprint, 2016, arXiv:1601.02470.Google Scholar
Amore, P., Fernández, F. M. and Hofmann, C. P., “Weakly bound states in heterogeneous waveguides”, Eur. Phys. J. B 89 (2016) 163; doi:10.1140/epjb/e2016-70197-0.CrossRefGoogle Scholar
Weideman, J. A. C. and Cloot, A., “Spectral methods and mappings for evolution equations on the infinite line”, Comput. Methods Appl. Mech. Eng. 80 (1990) 467–481; doi:10.1016/0045-7825(90)90052-N.Google Scholar
Avishai, Y., Bessis, D., Giraud, B. G. and Mantica, G., “Quantum bound states in open geometries”, Phys. Rev. B 44 (1991) 8028–8034; doi:10.1103/PhysRevB.44.8028.Google Scholar
Bittner, S., Dietz, B., Miski-Oglu, M., Richter, A., Ripp, C., Sadurní, E. and Schleich, W. P., “Bound states in sharply bent waveguides: analytical and experimental approach”, Phys. Rev. E 87 (2013) 042912; doi:10.1103/PhysRevE.87.042912.Google Scholar
Borisov, D., Exner, P., Gadyl’shin, R. and Krejčiřik, D., “Bound states in weakly deformed strips and layers”, Ann. Henri Poincaré 2 (2001) 553–572; doi:10.1007/PL00001045.Google Scholar
Boyd, J. P., “Spectral methods using rational basis functions on an infinite interval”, J. Comput. Phys. 69 (1987) 112–142; doi:10.1016/0021-9991(87)90158-6.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; doi:10.1090/S0002-9939-97-03726-X.Google Scholar
Carini, J. P., Londergan, J. T., Mullen, K. and Murdock, D. P., “Bound states and resonances in waveguides and quantum wires”, Phys. Rev. B 46 (1992) 15538; doi:10.1103/PhysRevB.46.15538.Google Scholar
Carini, J. P., Londergan, J. T., Mullen, K. and Murdock, D. P., “Multiple bound states in sharply bent waveguide”, Phys. Rev. B 48 (1993) 4503–4515; doi:10.1103/PhysRevB.48.4503.CrossRefGoogle 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; doi:10.1142/S0129055X95000062.Google Scholar
Exner, P. and Kovařík, H., Quantum waveguides (Springer, Cham, 2015); doi:10.1007/978-3-319-18576-7.Google Scholar
Gat, G. and Rosenstein, B., “New method for calculating binding energies in quantum mechanics and quantum field theories”, Phys. Rev. Lett. 70 (1993) 5–8; doi:10.1103/PhysRevLett.70.5.Google Scholar
Granot, E., “Emergence of a confined state in a weakly bent wire”, Phys. Rev. B 65 (2002) 233101; doi:10.1103/PhysRevB.65.233101.CrossRefGoogle Scholar
Hurt, N. E., Mathematical physics of quantum wires and devices: from spectral resonances to Anderson localization (Kluwer, Dordrecht, 2000); doi:10.1007/978-94-015-9626-8.CrossRefGoogle Scholar
Levin, D., “On the spectrum of the Dirichlet Laplacian on broken strips”, J. Phys. A 37 (2003) L9–L11; doi:10.1088/0305-4470/37/1/L02.Google Scholar
Londergan, J. T., Carini, J. P. and Murdock, D. P., Binding and scattering in two-dimensional systems: applications to quantum wires, waveguides and photonic crystals (Springer, Berlin, 1999); doi:10.1007/3-540-47937-6.Google Scholar
Nazarov, S. A., “Discrete spectrum of cranked, branching, and periodic waveguides”, St. Petersburg Math. J. 23 (2012) 351–379; doi:10.1090/S1061-0022-2012-01200-8.Google Scholar
Reed, M. and Simon, B., Methods of modern mathematical physics. IV. Analysis of operators (Academic Press, New York, 1978).Google Scholar
Sadurní, E. and Schleich, W. P., “Conformal mapping and bound states in bent waveguides”, AIP Conf. Proc. 1323 (2010) 283; doi:10.1063/1.3537857.Google Scholar
Simon, B., “The bound state of weakly coupled Schrödinger operators in one and two dimensions”, Ann. Phys. 97 (1976) 279–288; doi:10.1016/0003-4916(76)90038-5.Google Scholar
Tantau, T., “The TikZ and PGF Packages, Manual for version 3.0.0”; http://sourceforge.net/projects/pgf/.Google Scholar