Abstract A quasi two-dimensional periodic B-Si-B sandwich structure reveals surprising properties, including high-temperature superconductivity at 145 K0 [3]. This effect can be interpreted as superconductivity-gap (SC-gap) enhancing resulting from hybridisation of Bloch-functions of the weakly interacting Boron’s doped upper and low plates of the sandwich, resulting in the restructuring of systems of spectral band/gaps, similarly to [1]. In this paper we suggest a soft model of a two-dimensional periodic lattice and a quasi two-dimensional periodic sandwich and calculate the typical dispersion function of the objects based on a rational approximation of the corresponding Dirichlet-to-Neumann maps (DN-maps ). The rational approximations are interpreted as DN -maps of the corresponding fitted solvable models, which reveal some interesting physical properties such as Landau-Zener enhancing of the SC gap which may imply the high temperature superconductivity phenomenon.