A model of a quantum waveguide multiplexer





quantum waveguide, transmission, resonance, operator extensions theory, mesoscopic system


The paper explores the system of quantum waveguides and resonators. It suggests a solvable model of zero-width coupling windows based on the operator extensions theory in the Pontryagin space with indefinite metrics. The model self-adjoint operator is constructed explicitly, and it is similar (in some sense) to that of a physical system. We obtain an expression for the transmission coefficient for electrons and investigate its dependence on the electron energy which has a resonant character. It allows to control the electron transmission to different waveguides. The paper proposes a model of a three-channel quantum mesoscopic multiplexer.


Albeverio, S., Gesztesy, F., Hoegh-Krohn, R., Holden, H. (2005) Solvable models in quantum mechanics. 2nd ed. S. p.: AMS Chelsea Publishing, 850 p. (In English)

Beenakker, C. W. J., van Houten, H. (1991) Quantum transport in semiconductor nanostructures. Solid State Physics, 44, 1–228. DOI: 10.1016/S0081-1947(08)60091-0 (In English)

Buttiker, M. (1993) Capacitance, admittance, and rectification properties of small conductors. Journal of Physics: Condensed Matter, 5 (50), article 9361. DOI: 10.1088/0953-8984/5/50/017 (In English)

Derkach, V., Hassi, S., de Snoo, H. (2003) Singular perturbations of self-adjoint operators. Mathematical Physics, Analysis and Geometry, 6 (4), 349–384. DOI: 10.1023/B:MPAG.0000007189.09453.fc (In English)

Exner, P., Kovarik, H. (2015) Quantum waveguides. Berlin: Springer Publ., 422 p. DOI: 10.1007/978-3-319-18576-7 (In English)

Landauer, R. (1957) Spatial variation of currents and fields due to localized scatterers in metallic conduction. IBM Journal of research and development, 1 (3), 223–231. DOI: 10.1147/rd.13.0223 (In English)

Landauer, R. (1970) Electrical resistance of disordered one-dimensional lattices. The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics. Series 8, 21 (172), 863–867. DOI: 10.1080/14786437008238472 (In English)

Melikhova, A. S., Popov, I. Yu. (2017) Spectral problem for solvable model of bent nano peapod. Applicable Analysis, 96 (2), 215–224. DOI: 10.1080/00036811.2015.1120289 (In English)

Nazarov, S. A., Orive-Illera, R., Pérez-Martínez, M.-E. (2019) Asymptotic structure of the spectrum in a Dirichlet-strip with double periodic perforations. Networks & Heterogeneous Media, 14 (4), 733–757. DOI: 10.3934/nhm.2019029 (In English)

Pavlov, B. S. (1987) The theory of extensions and explicitly-soluble models. Russian Mathematical Surveys, 42 (6), 127–168. DOI: 10.1070/RM1987v042n06ABEH001491 (In English)

Pavlov, B. S., Popov, I. Yu., Frolov, S. V. (2001) Quantum switch based on coupled waveguides. The European Physical Journal B — Condensed Matter and Complex Systems, 21 (2), 283–287. DOI: 10.1007/s100510170203 (In English)

Popov, I. Yu. (1992) The resonator with narrow slit and the model based on the operator extensions theory. Journal of Mathematical Physics, 33 (11), 3794–3801. DOI: 10.1063/1.529877 (In English)

Popov, I. Yu. (1997) On the point and continuous spectra for coupled quantum waveguides and resonators. Reports on Mathematical Physics, 40 (3), 521–529. DOI: 10.1016/S0034-4877(97)85901-0 (In English)

Popov, I. Yu. (2013) Model of point-like window for electromagnetic Helmholtz resonator. Zeitschrift für Analysis und ihre Anwendungen, 32 (2), 155–162. DOI: 10.4171/ZAA/1478 (In English)

Popov, I. Yu., Popova, S. L. (1993a) The extension theory and resonances for a quantum waveguide. Physics Letters A, 173 (6), 484–488. DOI: 10.1016/0375-9601(93)90162-S (In English)

Popov, I. Yu., Popova, S. L. (1993b). Zero-width slit model and resonances in mesoscopic systems. Europhysics Letters, 24 (5), 373–373. DOI: 10.1209/0295-5075/24/5/009 (In English)

Shondin, Yu. G. (1988) Quantum-mechanical models in Rn associated with extensions of the energy operator in a Pontryagin space. Theoretical and Mathematical Physics, 74 (3), 220–230. DOI: 10.1007/BF01016615 (In English)

Sols, F., Macucci, M., Ravaioli, U., Hess, K. (1989) Theory for a quantum modulated transistor. Journal of Applied Physics, 66 (8), 3892–3906. DOI: 10.1063/1.344032 (In English)

Takagaki, Y., Ploog, K. (1994) Ballistic electron transmission in coupled parallel waveguides. Physical Review B, 49 (3), 1782–1788. DOI: 10.1103/PhysRevB.49.1782 (In English)

van Diejen, J. F., Tip, A. (1991) Scattering from generalized point interactions using self–adjoint extensions in Pontryagin spaces. Journal of Mathematical Physics, 32 (3), 630–641. DOI: 10.1063/1.529404 (In English)

Vorobiev, A. V., Bagmutov, A. S., Popov, A. I. (2019) On formal asymptotic expansion of resonance for quantum waveguide with perforated semitransparent barrier. Nanosystems: Physics, Chemistry, Mathematics, 10 (4), 415–419. DOI: 10.17586/2220-8054-2019-10-4-415-419 (In English)





Theoretical Physics