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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 $\alpha=1$., Comment: 5 pages

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., Comment: 15 pages, 32 figures

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-dimensional versions of the knights-like models., Comment: 13 pages, 18 figures; Spiral model does satisfy property B

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions., Comment: 35 pages, 24 figures. Improved version with one added figure (figure 9), a shift s->s+1 in the definition of the tree size, and minor corrections

Toninelli, Biroli, and Fisher recently introduced the knights model, a correlated percolation system (Phys. Rev. Lett. 96, 035702 (2006)). They claimed to prove that the critical point of this model was the same as that for directed percolation, and then used this to show (assuming a conjecture about directed percolation, described later in a more detailed proof) that this model has a discontinuous phase transition with a diverging correlation length. However, there is an error in their work, so that these results are unproven for their model. Their proofs can, however, be modified to work for a similar model., Comment: 1 page, 1 figure

We calculate all multipoint correlation functions of all local bond modifications in the two-dimensional Abelian sandpile model, both at the critical point, and in the model with dissipation. The set of local bond modifications includes, as the most physically interesting case, all weakly allowed cluster variables. The correlation functions show that all local bond modifications have scaling dimension two, and can be written as linear combinations of operators in the central charge -2 logarithmic conformal field theory, in agreement with a form conjectured earlier by Mahieu and Ruelle in Phys. Rev. E 64, 066130 (2001). We find closed form expressions for the coefficients of the operators, and describe methods that allow their rapid calculation. We determine the fields associated with adding or removing bonds, both in the bulk, and along open and closed boundaries; some bond defects have scaling dimension two, while others have scaling dimension four. We also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries., Comment: 13 pages, 5 figures; referee comments incorporated; Accepted by Phys. Rev. E

We study how quasiparticle interactions affect their localization properties in dirty superconductors with broken time reversal symmetry -- for example in a magnetic field. For SU(2) spin-rotation invariant (class C) systems, the only important coupling is the spin-spin triplet interaction, which we study within a renormalization group approach. Either an additional Zeeman coupling or a complete breaking of spin rotation symmetry renders all interactions irrelevant. These two situations realize, respectively, the non-interacting unitary Anderson and the ``thermal'' (class D) universality class. Our results imply a stable metallic phase in 2D for class D. Experimental implications are discussed., Comment: 5 pages, Bull. Am. Phys. Soc. 46, 231 (2001)

We consider an ideal gas of infinitely rigid rods near a perfectly repulsive wall, and show that the interfacial tension of a surface with rods on one side is lower when the surface bends towards the rods. Surprisingly we find that rods on both sides of surfaces also lower the energy when the surface bends. We compute the partition functions of rods confined to spherical and cylindrical open shells, and conclude that spherical shells repel rods, whereas cylindrical shells (for thickness of the shell on the order of the rod-length) attract them. The role of flexibility is investigated by considering chains composed of two rigid segments., Comment: 39 pages including figures and tables. 12 eps figures. LaTeX with REVTeX

Recent laboratory experiments have measured fusion cross sections at center-of-mass energies low enough for the effects of atomic and molecular electrons to be important. To extract the cross section for bare nuclei from these data (as required for astrophysical applications), it is necessary to understand these screening effects. We study electron screening effects in the low-energy collisions of Z=1 nuclei with hydrogen molecules. Our model is based on a dynamical evolution of the electron wavefunctions within the TDHF scheme, while the motion of the nuclei is treated classically. We find that at the currently accessible energies the screening effects depend strongly on the molecular orientation. The screening is found to be larger for molecular targets than for atomic targets, due to the reflection symmetry in the latter. The results agree fairly well with data measured for deuteron collisions on molecular deuterium and tritium targets., Comment: 15 Page RevTeX document, twelve postscript figures, now in a uufile package