On the nonlocality of the fractional Schr{o}dinger equation

A number of papers over the past eight years have claimed to solve the fractional Schr{o}dinger equation for systems ranging from the one-dimensional infinite square well to the Coulomb potential to one-dimensional scattering with a rectangular barrier. However, some of the claimed solutions ignore the fact that the fractional diffusion operator is inherently nonlocal, preventing the fractional Schr{o}dinger equation from being solved in the usual piecewise fashion. We focus on the one-dimensional infinite square well and show that the purported groundstate, which is based on a piecewise approach, is definitely not a solution of the fractional Schr{o}dinger equation for general fractional parameters $alpha$. On a more positive note, we present a solution to the fractional Schr{o}dinger equation for the one-dimensional harmonic oscillator with $alpha=1$.

On the study of force-balance percolation

We study models of correlated percolation where there are constraints on the occupation of sites that mimic force-balance, i.e. for a site to be stable requires occupied neighboring sites in all four compass directions in two dimensions. We prove rigorously that $p_c<1$ for the two-dimensional models studied. Numerical data indicate that the force-balance percolation transition is discontinuous with a growing crossover length, with perhaps the same form as the jamming percolation models, suggesting the same underlying mechanism driving the transition in both cases. In other words, force-balance percolation and jamming percolation may indeed belong to the same universality class. We find a lower bound for the correlation length in the connected phase and that the correlation function does not appear to be a power law at the transition. Finally, we study the dynamics of the culling procedure invoked to obtain the force-balance configurations and find a dynamical exponent similar to that found in sandpile models.

On the study of jamming percolation

We investigate kinetically constrained models of glassy transitions, and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and time scales. The models we investigate have constraints similar to that of the knights model, introduced by Toninelli, Biroli, and Fisher (TBF), but differing neighbor relations. We find that such knights-like models, otherwise known as models of jamming percolation, need a ``No Parallel Crossing rule for the TBF proof of a glassy transition to be valid. Furthermore, most knight-like models fail a ``No Perpendicular Crossing requirement, and thus need modification to be made rigorous. We also show how the ``No Parallel Crossing requirement can be used to evaluate the provable glassiness of other correlated percolation models, by looking at models with more stable directions than the knights model. Finally, we show that the TBF proof does not generalize in any straightforward fashion for three-dimensiona