Georgia Institute of Technology College of Sciences

Yves Couder and coworkers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit wave-particle duality and, as a consequence, several dynamical features previously thought to be peculiar to the microscopic realm, including single-particle diffraction, interference, tunneling and quantized orbits. We explore this fluid system in light of the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. Doing so reveals a remarkable correspondence between bouncing droplets and subatomic particles, and provides rationale for the observed macroscopic quantum behaviour. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.

Suppose that x(t) is a signal generated by a chaotic system and that the signal has been recorded in the interval [0,T]. We ask: What is the largest value t_f such that the signal can be predicted in the interval (T,T+t_f] using the history of the signal and nothing more? We show that the answer to this question is contained in a major result of modern information theory proved by Wyner, Ziv, Ornstein, and Weiss. All current algorithms for predicting chaotic series assume that if a pattern of events in some interval in the past is similar to the pattern of events leading up to the present moment, the pattern from the past can be used to predict the chaotic signal. Unfortunately, this intuitively reasonable idea is fundamentally deficient and all current predictors fall well short of the Wyner-Ziv bound. We explain why the current methods are deficient and develop some ideas for deriving an optimal predictor. [This talk is based on joint work with X. Liang and K. Serkh]. To view and/or participate in the webinar from wherever you are, click on:EVO.caltech.edu/evoNext/koala.jnlp?meeting=MvM2Ml2M2tDvDn9n9nDe9v

We study a two dimensional Schrödinger operator for a limit-periodic potential. We prove that the spectrum contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves in the high energy region. Second, the isoenergetic curves in the space of momenta corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). Third, the spectrum corresponding to these eigenfunctions (the semiaxis) is absolutely continuous.

An explicit asymptotic expression for the ground-state energy of the Pekar-Tomasevich functional for the N-polaron is found, when the positive repulsion parameter U of the electrons is less than twice the coupling constant of the polaron. This is joint workwith Gonzalo Bley.