Multidimensional Periodic Schrodinger Operator Perturbation Theory and Applications Conclusions

Abstract

In this chapter we summarize the results and methods of the book in the descriptive way. In this book the following three problems of the spectral theory of the multidimensional Schrodinger operator L(q) with a periodic potential q are investigated. 1. First problem is the perturbation theory of L(q) (Chap. 2). 2. Second problem is the constructive determination of the spectral invariants of L(q) from the given Bloch eigenvalues (Chap. 3). 3. Third problem is the constructive determination of the potential q from the spectral invariants (Chap. 4). We describe all these three problems that can be considered as unique features. Moreover these problems are investigated as a whole and in the pertinent form in the sense that First problem. Second problem. Third problem. Besides in Chap. 1 we present some definitions, statements and discussions to be used in the next chapters from the point of view of both the physicists and mathematicians.