Integral kernels of the scattering matrices for time-periodic Schrödinger equations (Q1107039)

The author considers the time-dependent Schrödinger equation (1) \(iu_ t=-\Delta u+V(x,t)u\) where V is a time periodic (real) potential and shows some important results concerning the kernel of the scattering matrix associated with problem (1) such as smoothness (off-diagonal) and decay properties. His additional assumptions on V are: a) \(V\in C^{\infty}\) and b) There exist \(\delta >0\) with \(<\delta <3/2\) and \(C>0\) such that for any \(\alpha \in {\mathbb{N}}^ n\) and \(k\in {\mathbb{N}}\) we have that \[ | D_ x^{\alpha}(\partial /\partial t)^ kV(x,t)| \leq C(\alpha,k)(1+| x|^ 2)^{-(2\delta +| \alpha |)}. \] The presentation of the subject is concise, mathematically rigorous and yet very intelligible and lucid. The considerations are illustrated by many examples, whilst the proportions between theory and applications are weighted excellently; also exercises after every section and references after each chapter are included. The book should be easily adapted and very useful for graduate students; it may serve too as a teaching aid for lecturers at universities and a source of reference for research workers.