During 2014 our usage of cluster Zefiro (group dynsysmath) has been less than expected because it has been employed mostly for refined calculation of a long review paper on the quantum dynamics of cold atoms (Bose condensates) in optical lattices, and for preliminary investigations of a spectral problem for almost periodic Hamiltonians.
The review paper , written by myself with Andrey Kolosvsky of the Kirensky Institute of Physics of the Siberian Academy of Sciences, has been pre-accepted as a candidate "Physics Reports" and is currently under the scrutiny of the editor, David Campbell. Its title is "Landau-Stark states and cyclotron Bloch oscillations of quantum particles in two-dimensional lattices", and in six main chapters and 40 pages it reviews existing literature and our papers on the problem. We have described theoretical techniques to solve the quantum spectral problem of a particle under the combined action of a periodic potential, an electric field and a magnetic field in the Hall configuration. At difference with existing approaches, neither of these fields is treated perturbatively. This is required by the present application of the theory to Bose condensates, mentioned above. Our theory is always accompanied by numerical simulations. While in some cases they are rather simple numerically–albeit non trivial theoretically–in others they required the usage of the cluster. This is particularly true for the numerical computations of the quantum dynamics of wave-packets, coherent and incoherent, that simulate the results of real laboratory experiments. The latter case has required the simulation of an ensemble of "phase scrambled" wave-packets, that would not have been possible without the usage of the cluster. Yet, since the theory has been already well established and the numerical codes tested, the cpu usage employed has been comparatively low: our effort this year has been mainly towards the redaction of this long review. It was not our original plan, but given the prestige of "Physics Reports" we decided that it was worth to try.
In 2015 we would like to come back to what was originally planned for 2014, that is, an extended investigation of the spectral and dynamical properties of almost periodic one–dimensional and many–dimensional tight binding systems, with applications to the physical systems described above (quantum condensates in optical lattices) as well to chains of classical oscillators. Here, the theory is much more difficult than in the periodic case, because it requires deep concepts of harmonic analysis and of the logarithmic potential theory in the complex plane. I have started investigating this problem in 2014, with preliminary runs on the cluster, and two papers describing the first results [2,3] are currently under review. In these papers, I show how parallel computing is fundamental to solve complicated equations for a kind of electrostatic potential generated by a charge supported on a fractal set. At difference with the "conventional" physical case, the repulsion law of the charge is here logarithmic, as it were created by infinite charged wires perpendicular to a plane. In 2015 I would like to pursue this problem at depth, and I therefore would ask again for 50,000 core hours.
1) A. Kolovsky, G. Mantica, "Landau-Stark states and cyclotron Bloch oscillations of quantum particles in two dimensional lattices", submitted to Physics Reports.
2) G. Mantica, "Orthogonal polynomials of equilibrium measures supported on Cantor sets", submitted to Journal of Computational and Applied Mathematics.
3) G. Mantica, "The isospectral torus of finite gap sets and of Cantor sets", submitted to Computational Methods and Function Theory.