Abstract
In this document we review some results dealing with the study of the spectral properties of quantum waveguide. Precisely we are interested in the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width d. We impose the Neumann boundary condition on a disc window of radius a and Dirichlet boundary conditions on the remained part of the boundary of the strip (Najar et al., Math Phys Anal Geom 13:19–28, 2010).
We study the case when we destroy the plan symmetry, i.e. we impose the Neumann boundary condition on the two concentric disc windows of the radii a and b located on the opposite walls and the Dirichlet boundary condition on the remaining part of the boundary of the strip (Najar and Olendski, J Phys A Math Theor 44, 2011).
The effect of a magnetic field of Aharonov-Bohm type when the magnetic field is turned on this system is considered (Najar and Raissi, On the spectrum of the Schrodinger Operator with Aharonov-Bohm Magnetic Field in quantum waveguide with Neumann window, Math. Meth. App. Sci. (2015)).
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Najar, H. (2015). Spectral results on quantum waveguides. In: Jeribi, A., Hammami, M., Masmoudi, A. (eds) Applied Mathematics in Tunisia. Springer Proceedings in Mathematics & Statistics, vol 131. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18041-0_4
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