| Abstract: | Up until the mid 70s the kind of spectra most people had in mind in the context of theory of Schrodinger operators were spectra occurring for periodic potentials and for atomic and molecular Hamiltonians. Then evidence started to build up that "exotic" spectral phenomena such as singular continuous, Cantor, and dense point spectrum do occur in mathematical models that are of substantial interest to theoretical physics. One area where such exotic phenomena are particularly abundant is quasiperiodic operators. They feature a competition between randomness (ergodicity) and order (periodicity), which is often resolved at a deep arithmetic level. Mathematically, the methods involved include a mixture of ergodic theory, dynamical systems, probability, functional and harmonic analysis. The interest in those models was enhanced by strong connections with some major discoveries in physics, such as integer quantum Hall effect, experimental quasicrystals, and quantum chaos theory, in all of which quasiperiodic operators provide central or important models. We will give a general overview concentrating on aspects where the competition and/or collaboration between order and chaos plays an important role, and then will overview recent results on sharp arithmetic transitions for explicit physics-central models. Those models all demonstrate interesting dependence on the arithmetics of parameters (even in some cases when the final conclusion does not have such dependence) and have traditionally been approached through KAM-type schemes. Even when the KAM arguments have been replaced by the non-perturbative ones allowing to treat more couplings, frequencies that are neither far from nor close enough to rationals presented a challenge as for them there was nothing left to perturb about. A remarkable relatively recent development concerning the explicit models is that very precise results have become possible: not only many facts have been established for a.e. frequencies and phases, but in many cases it has become possible to go deeper in the arithmetics and either establish precise arithmetic transitions or even obtain results for all values of parameters. |
