Landau–Zener effect for a quasi-2D periodic sandwich

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.