Perturbation Theory for the Schrödinger Operator with a Periodic Potential The book is devoted to perturbation theory for the Schrödinger operator with a periodic potential describing motion of a particle in bulk matter. The Bloch eigenvalues of the operator are densely situated in a high energy region so regular perturbation theory is ineffective. The mathematical difficulties have a physical nature a complicated picture of diffraction inside the crystal. The author develops a new mathematical approach to this problem. It provides mathematical physicists with important results for this operator and a new technique that can be effective for other problems. The semiperiodic Schrödinger operator describing a crystal with a surface is studied. Solid body theory specialists can find asymptotic formulae which are necessary for calculating many physical values