Geodesic
Landau--Zener effect for a quasi-2D periodic sandwich
Nanosistemy: fizika, himiâ, matematika, Tome 2 (2011) no. 4, pp. 32-50.

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Bloch-waves in 1D periodic lattices are typically constructed based on the transfer-matrix approach, with a complete system of solutions of the Cauchy problem on a period. This approach fails for the multidimensional Schrödinger equations on periodic lattices, because the Cauchy problem is ill-posed for the associated elliptic partial differential equations. In our previous work we suggested a different procedure for the calculation of the Bloch functions for the 2D Schrödinger equation based on the Dirichlet-to-Neumann map substituted for the transfer-matrix. In this paper we suggest a method of calculation of the dispersion function and Bloch waves of quasi-2D periodic lattices, in particular of a quasi-2D sandwich, based on construction of a fitted solvable model.
Keywords: landau–Zener effect, bloch waves. 
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     author = {N. Bagraev and G. Martin and B. S. Pavlov and A. Yafyasov},
     title = {Landau--Zener effect for a {quasi-2D} periodic sandwich},
     journal = {Nanosistemy: fizika, himi\^a, matematika},
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     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/NANO_2011_2_4_a2/}
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N. Bagraev; G. Martin; B. S. Pavlov; A. Yafyasov. Landau--Zener effect for a quasi-2D periodic sandwich. Nanosistemy: fizika, himiâ, matematika, Tome 2 (2011) no. 4, pp. 32-50. http://geodesic.mathdoc.fr/item/NANO_2011_2_4_a2/