Perturbation theory for the Schrödinger operator with a periodic potential / Yulia E. Karpeshina

The main aim of this book is to construct perturbation formulae for Bloch eigenvalues and their spectral projections in a high energy region on a rich set of quasimomenta. The construction of these formulae is connected with the investigation of a complicated picture of the crystal diffraction. Another problem considered here is a semibounded crystal problem, i.e., the Schrödinger operator which has a zero potential in a half space and a periodic potential in the other half space. The interaction of a plane wave with a semicrystal is studied. First, the asymptotic expansion of the reflection coefficients in a high energy region is obtained, this expansion is valid for a rich set of momenta of the incident plane wave. Second, the connection of the asymptotic coefficients with the potential is established. Based upon these, the inverse problem is solved, this problem is to determine the potential from the asymptotics of the reflection coefficients in a high energy region (a crystallography problem). ... (Zentralblatt)