Geodesic
The discrete Schr\"odinger equation for a~quantum waveguide
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 80-93

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We investigate the spectral properties of the discrete Schrödinger operator for the infinite band with zero boundary conditions. We prove that the eigenvalues and resonances arise for the small decreasing potentials near singularities of the non-perturbed Green function (boundary points of the subbands) and we find their asymptotic behavior. The scattering picture is described: the diffraction (i.e. the scattering mainly in the finite number of preferential directions) transforms into probability waves in time of the reflection and propagation in the considered quasi-1D system. The simple formulas for these probabilities are obtained near boundary points of the subbands (this corresponds to small velocities of the quantum particles) for the small potentials.
Keywords: discrete Schrödinger operator, quantum waveguide, eigenvalue, resonance
Mots-clés : transmission and reflection coefficients. 
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     author = {T. S. Tinyukova and Yu. P. Chuburin},
     title = {The discrete {Schr\"odinger} equation for a~quantum waveguide},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {80--93},
     publisher = {mathdoc},
     number = {4},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2012_4_a6/}
}
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T. S. Tinyukova; Yu. P. Chuburin. The discrete Schr\"odinger equation for a~quantum waveguide. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 4 (2012), pp. 80-93. http://geodesic.mathdoc.fr/item/VUU_2012_4_a6/