On the discrete spectrum of a quantum waveguide with Neumann windows in presence of exterior field
https://doi.org/10.17586/2220-8054-2022-13-2-156-163
Abstract
The discrete spectrum of the Hamiltonian describing a quantum particle living in three dimensional straight layer of width d in the presence of a constant electric field of strength F is studied. The Neumann boundary conditions are imposed on a finite set of bounded domains (windows) posed at one of the boundary planes and the Dirichlet boundary conditions on the remaining part of the boundary (it is a reduced problem for two identical coupled layers with symmetric electric field). It is proved that such system has eigenvalues below the lower bound of the essential spectrum for any F ≥ 0. Then we closer examine a dependence of bound state energies on F and window’s parameters, using numerical methods.
About the Authors
A. S. Bagmutov
ITMO University
Russian Federation
Russian Federation
H. . Najar
Facultedes Sciences de Moanstir
Russian Federation
Russian Federation
I. F. Melikhov
ITMO University
Russian Federation
Russian Federation
I. Y. Popov
ITMO University
Russian Federation
Russian Federation
References
1. Borisov D., Exner P. and Gadyl’shin R., Krejcirik D. Bound states in weakly deformed strips and layers. Annales Henri Poincare´, 2001, 2(3), P. 553-572.
2. Briet Ph., Kovarik H., Raikov G. and Soccorsi E. Eigenvalue asymptotics in a twisted waveguide, Comm. PDE, 2009, 34(8), P. 818-836.
3. Bulla W. and Renger W. Existence of bound states in quantum waveguides under weak conditions. Lett. Math. Phys., 1995, 35(1), 1-12.
4. Chenaud B., Duclos P., Freitas P. and Krejcirik D. Geometrically induced discrete spectrum in curved tubes. Diff. Geom. Appl., 2005, 23(2), P. 95-105.
5. Duclos P. and Exner P. Curvature-induced bound state in quantum waveguides in two and three dimensions. Rev. Math. Phys., 1995, 7(1), P. 73-102.
6. Duclos P., Exner P. and Stovicek P. Curvature-induced resonances in a two-dimensional Dirichlet tube. Annales Henri Poincare´, 1995, 62(1), P. 81-101.
7. Ekholm T., Kovarik H. and Krejcirik D. A Hardy inequality in twisted waveguides. Arch. Rat. Mech. Anal., 2008, 188(2), P. 245-264.
8. Exner P. and Vugalter S.A. Bound States in a Locally Deformed Waveguide: The Critical Case. Lett. Math. Phys., 1997, 39(1)), P. 59-68.
9. Borisov D., Ekholm T. and Kovarik H. Spectrum of the magnetic Schrodinger operator in a waveguide with combined boundary conditions. Annales Henri Poincare´, 2005, 6(2), P. 327-342.
10. Ekholm T. and Kovarik H. Stability of the magnetic Schrodinger operator in a waveguide.Comm. PDE, 2005, 30(4), P. 539-565.
11. Grushin V.V. On the eigenvalues of finitely perturbed laplace operators in infinite cylindrical domains. Math. Notes, 2004, 75(3), P. 331-340.
12. Gadylshin R. On regular and singular perturbations of acoustic and quantum waveguides. C.R. Me´canique, 2004, 332(8), P. 647-652.
13. Bagmutov A.S., Popov I.Y. Window-coupled nanolayers: window shape influence on one-particle and two-particle eigenstates. Nanosystems: Phys. Chem. Math., 2020, 11(6), P. 636-641.
14. Melikhov I.F. and Popov I.Yu. Hartree-Fock approximation for the problem of particle storage in deformed nanolayer. Nanosystems: Physics, Chemistry, Mathematics, 2013, 4(4), P. 559-563.
15. Melikhov I.F. and Popov I.Yu. Multi-Particle Bound States in Window-Coupled 2D Quantum Waveguides. Chin. J. Phys., 2015, 53, 060802.
16. Popov S.I., Gavrilov M.I., and Popov I.Yu. Two interacting particles in deformed nanolayer: discrete spectrum and particle storage. Phys. Scripta, 2012, 86(3), 035003.
17. Zaremba S. Sur un proble´me toujours possible comprenant, a´ titre de cas particulier, le proble´me de Dirichlet et celui de Neumann. J. Math. Pure Appl., 1927, 6, P. 127-163.
18. Borisov D. Discrete spectrum of a pair of asymmetric window-coupled waveguides. English transl. Sb. Math., 2006, 197(3-4), P. 475-504.
19. Borisov D. On the spectrum of two quantum layers coupled by a window. J. Phys. A: Math. Theor., 2007, 40(19), P. 5045-5066.
20. Borisov D. and Exner P. Exponential splitting of bound states in a waveguide with a pair of distant windows. J. Phys A: Math and General, 2004, 37(10), P. 3411-3428.
21. Borisov D., Exner P. and Gadyl’shin R. Geometric coupling thresholds in a two-dimensional strip. J. Math. Phys., 2002, 43(12), P. 6265-6278.
22. Exner P. and Vugalter S.A. Bound state asymptotic estimate for window-coupled Dirichlet strips and layers. J. Phys. A: Math. Gen., 1997, 30(22), P. 7863-7878.
23. Exner P., Vugalter S.A. On the number of particles that a curved quantum waveguide can bind. J. Math. Phys., 1999, 40, P. 4630-4638
24. Popov I.Yu. Asymptotics of bound states and bands for laterally coupled three-dimensional waveguides. Rep. on Math. Phys., 2001, 48(3), P. 277-288.
25. Popov I.Yu. Asymptotics of bound state for laterally coupled waveguides. Rep. on Math. Phys., 1999, 43(3), P. 427-437.
26. Bulla W., Gesztesy F., Renger W. and Simon B. Weakly coupled bound states in quantum waveguides. Proc. Amer. Math. Soc, 1997, 125(5), P. 1487-1495.
27. Exner P., Sˇeba P., Tater M. and Vaneˇk D. Bound states and scattering in quantum waveguides coupled laterally through a boundary window. J. Math. Phys., 1996, 37(10), P. 4867-4887.
28. Najar H., S. Ben Hariz and M. Ben Salah. On the Discrete Spectrum of a Spatial Quantum Waveguide with a Disc Window. Math. Phys. Anal. Geom., 2010, 13(1), P. 19-28.
29. Najar H. and Olendski O. Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs. J. Phys. A: Math. Theor., 2011, 44(30), 305304.
30. Najar H. and Raissi M. A quantum waveguide with Aharonov-Bohm Magnetic Field. Math. Meth. App. Sci, 2016, 39(1), P. 92-103.
31. Popov I.Y., Bagmutov A.S., Melikhov I.F., Najar H. Numerical analysis of multi-particle states in coupled nano-layers in electric field. AIP Conf. Proc. Vol., 2020, 2293, 360006.
32. Numerical results a quantum waveguide with Mixed boundary conditions M Raissi - arXiv preprint arXiv:1501.02073, 2015 - arxiv.org
33. Dunne G. and Jaffe R.L. Bound states in twisted Aharonov-Bohm tubes. Ann.Phys., 1993, 233(2), P. 180-196.
34. Exner P. A Quantum Pipette. J. Phys A: Math and General, 1995, 28(18), P. 5323-5330.
35. Reed M. and Simon B. Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators. Academic Press, New York, 1978.
36. Reed M. and Simon B. Methods of Modern Mathematical Physics, Vol. I: Functionnal Analysis. Academic Press Inc; 2nd Revised edition, Academic Press, New York, 1981.
37. Wolf F. On the invariance of the essential spectrum under a change of boundary conditions of partial differential boundary operators. Indag. Math., 1959, 21, P. 142-147.
38. Goldstone J. and Jaffe R.L. Bound States in Twisting tubes. Phys. Rev. B, 1992, 45, 14100
Review
For citations:
Bagmutov A.S., Najar H., Melikhov I.F., Popov I.Y. On the discrete spectrum of a quantum waveguide with Neumann windows in presence of exterior field. Nanosystems: Physics, Chemistry, Mathematics. 2022;13(2):156-163. https://doi.org/10.17586/2220-8054-2022-13-2-156-163
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