The polaron concept was introduced by Landau in 1933 in the attempt to explain the optical properties of alkali halides in terms of electrons self-trapped by induced lattice polarization. Ever since then,the polaron problem (dealing with a single electron interacting with the lattice distortions in a crystal) has attracted the attention of researchers. The early works on polarons concerned the theoretical formulation of this problem in polar and ionic compounds as well as metals. In particular,the lattice was treated within the continuum approximation (the spatial extension of the lattice distortion being large compared to the size of the unit atomic cell) and the lattice dynamics was described within the dielectric formulation. The first important outcome of this polaron physics, due primarily to H. Fr¨ohlich,w as related to the phenomenon of superconductivity. He was the first to suggest that it was caused by an electron-phonon interaction. He was able to explain the isotope effect of these compounds which ultimately led to the B.C.S. theory,based on lattice-mediated electron pairing. By the 1960s the polaron features in the continuum approximation had been largely understood for weak electron-phonon coupling strengths,alb eit as regards the single-polaron problem. The question which became more and more important to be answered at that time was whether,b y increasing the electron-lattice coupling strength,a transition would occur when going from the large itinerant Fr¨ohlich polaron (where the lattice distortion is large compared to the interatomic spacing) to the small polaron (where the lattice distortion size is about the size of the atomic unit cell). It became clear very early on that a very abrupt cross-over must occur in such systems,without however implying a true phase transition. The attention was thus subsequently directed to the study of lattice polarons. The Fr¨ohlich Hamiltonian for continuum systems was adapted in order to treat the motion of electrons on a lattice with local coupling to deformable molecular units,leading to the so-called Holstein molecular crystal model.

In the last thirty years or so an enormous theoretical effort was made to treat electronphonon coupled systems,with particular emphasis on Many-Body aspects for dense electron systems,taking into account continuum as well as lattice polaron effects. Features which characterize such systems and which have been extensively dealt with in this School concern among others: the mutual attractive interaction between polarons giving rise to non–symmetry-breaking metal-insulator transitions,symmetry-breaking metal-charge density wave insulator transitions,bip olaronic superconductivity and resonant pairing, electron-density–driven polarization catastrophes, Wigner crystallization in systems with long-range Coulomb interaction and the influence of strong correlations as well as orbital degeneracy on the effectiveness of the electron-phonon interaction.

Treating such aspects of polaron theory has been made possible because of powerful Many Body techniques which have been developed over the past few decades and which include: Exact Diagonalization techniques,Quan tum Monte Carlo approaches,Densit y Matrix renormalization group and Dynamical Mean-Field Theory. All these advances in polaron theory would have been unthinkable had they not been accompanied by: i) an equally important advance in material research which produced many new materials such as the high-T_c cuprates,the manganites and nickelates and the fullerines; ii) as well as significant advances in the refinement of experimental analysis and,in particular, the spectroscopic means such as Angle-Resolved Photoemission Spectroscopy,X-Ra y Absorption Spectroscopy (EXAFS,XANES),Pulsed Neutron Diffraction measurements allowing to study the local dynamical lattice deformations and optical spectroscopy including time-resolved measurements.

The scope and purpose of this School was to review both these theoretical and experimental advances which occurred over the last few decades and introduce students to the study of such systems,where both strong electron-electron correlations and large electron-phonon coupling strengths play important roles. The free time reserved in the afternoons permitted and stimulated a lively exchange of ideas between the participants, lecturers as well as students.

We thank the Italian Physical Society (SIF) for financial and logistical support. We thank also the SIF Secretariat and the Local Organization as well as the Editorial Staff. Their help allowed the publication of these Proceedings. We thank all the lecturers for the diligence in the preparation of their talks,the preparation of their manuscripts for the present volume as well as for their continued interaction with the students during this School. We thank the Consiglio Nazionale delle Ricerche (CNR),the Istituto Nazionale di Fisica Nucleare (INFN),the Center Coherentia at the Napoli Unit of the CNR-INFM (Istituto Nazionale per la Fisica della Materia) and the Gruppo di Napoli di Geofisica at the Physical Sciences Department of the University of Napoli “Federico II” for their sponsorship and financial support.

G. IADONISI, J. RANNINGER AND G. DE FILIPPIS