Local dispersion analysis for periodic lattices

Inverse Problems and Applications

15 April 14:00 - 15:00


Due to Fermi-Dirac statistics of electrons in solid the nontrivial dynamics of electrons for low temperature is observed only locally on a small temperature interval of energy near the Fermi level. The dispersion defines properties of electrons from the Bloch - functions which are constructed in one-dimensional case from standard solutions of the Cauchy problem on the period. This approach fails for multi-dimensional Schrödinger lattices, because the corresponding Cauchy problem is ill posed. We suggest to construct the Bloch waves for multi-dimensional periodic Schrödinger lattices from solutions of standard boundary problems, based on the DN-map. The local dynamics and spectral properties are defined by the local dispersion recovered based on the rational approximation of the DN-map. Further simplification of the direct problem of local spectral analysis is achieved when the essential localization of electrons on deep spectral bands inside the period is taken into account. It can be done by selecting appropriate finite-dimensional contact spaces on the boundary of the period and substitution of the DN-map on the period by the partial DN-map obtained via framing the original one by projections onto the contact subspaces. The partial DN-map plays a role of the DN-map of the the corresponding model lattice, which has similar spectral spectral properties, with chemical bonds defined by matching in contact spaces. The local description of basic spectral properties of the periodic lattice near the Fermi level can be considered as a solution of the direct problem of the local spectral analysis. Vice versa, construction of the fitted solvable model of the lattice , with prescribed local spectral and dynamical properties, can be interpreted as a solution of the inverse local problem.