In many physical situations, separation of scales plays a fundamental role in
understanding the dynamical behavior of the system. In particular, we focus
on physical systems in which it is possible to distinguish between fast and slow
degrees of freedom. The goal is to obtain an effective Schrodinger equation governing
the dynamics of the slow degrees of freedom, thereby greatly simplifying
the complexity of the problem.
The previous lines summarize the spirit and the goals of the theory outlined
in this thesis. The thesis collects original results obtained as a joint work with
Herbert Spohn and Stefan Teufel, who initiated this research project some years
before and introduced me into this field of research. The results have been obtained
during the second part of my Ph. D. studies at SISSA, Trieste, under the
internal supervision of Gianfausto Dell' Antonio.
Since the reader will be probably looking forward to read the main body of
the thesis, I will spend just few more words about the novelty of the results and
the references to the literature.
As far as the novelty of the results is concerned, all the results appearing in
the main body of the thesis are essentially new, with the exception of Egorov's
theorem in Ch. 2 and few minor propositions. As opposed, the results reviewed
in the Appendix appeared already in the literature, see for example [Ho, Fo, Iv,
GMS].
Detailed references to the literature and to related approaches will be given
sectionwise, so that the comparison of the methods and the results will be easier.
However, I wish to mention here that the results in [Em We, NeSo] have been
greatly inspiring for us.
At the risk of being pedantic, I wish to emphasize that all the results should
be considered as the fruit of a joint work with Herbert Spohn and Stefan Teufel,
although this will not be explicitly mentioned sectionwise.
The introductory chapter looks very much as the transcription of the talk I
had the occasion to give in many places (Vienna, Taxco, Trieste, Rome, Cala
Gonone, Bielefeld, ... ) in the last months. Indeed it is. But this is a consequence
of the precise choice to make the first chapter as readable as possible, so that I avoided any use of technical concepts in the Introduction. This is also the reason
why references to the literature do not appear in the introduction.
Finally, I took for myself the freedom to break a very solid convenction. Indeed,
in this thesis the word "hamiltonian" is written without the capital letter,
since in the last eight years - since my first course in rational mechanics - no body
was able to explain to me why the words "algebraic", "bosonic", "euclidean" or
"fermionic" are usually written in small letters, while "hamiltonian" should be
promoted to the capital letter. I hope that Sir Hamilton will not be too much
offended for that and, more important, will not take this fact too seriously.
