A Conservative Scheme with Optimal Error Estimates for a Multidimensional Space-Fractional Gross-Pitaevskii Equation

The present work departs from an extended form of the classical multi-dimensional Gross-Pitaevskii equation, which considers fractional derivatives of the Riesz type in space, a generalized potential function and angular momentum rotation. It is well known that the classical system possesses functionals which are preserved throughout time. It is easy to check that the generalized fractional model considered in this work also possesses conserved quantities, whence the development of conservative and efficient numerical schemes is pragmatically justified. Motivated by these facts, we propose a finite-difference method based on weighted-shifted Grünwald differences to approximate the solutions of the generalized Gross-Pitaevskii system. We provide here a discrete extension of the uniform Sobolev inequality to multiple dimensions, and show that the proposed method is capable of preserving discrete forms of the mass and the energy of the model. Moreover, we establish thoroughly the stability and the convergence of the technique, and provide some illustrative simulations to show that the method is capable of preserving the total mass and the total energy of the generalized system. © 2019 Ahmed S. Hendy et al., published by Sciendo 2019

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Symmetries in Quantum Mechanics and Statistical Physics

This book collects contributions to the Special Issue entitled "Symmetries in Quantum Mechanics and Statistical Physics" of the journal Symmetry. These contributions focus on recent advancements in the study of PT–invariance of non-Hermitian Hamiltonians, the supersymmetric quantum mechanics of relativistic and non-relativisitc systems, duality transformations for power–law potentials and conformal transformations. New aspects on the spreading of wave packets are also discussed

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

The evolution of spherically symmetric configurations in the Schrödinger equation approach to cosmic structure formation

The evolution of spherically symmetric cold dark matter overdensities in an expanding Universe is studied using Schrödinger-Newton (SN) equations, which model self-gravitating collisionless matter. For doing so, the density profiles of the perturbations are ideally divided into shells, for which an explicit SN solution for a Λ=0 background can be found. Then, supposing absence of shell crossing during the whole evolution of the overdensity, the free-particle approximation is applied to each shell. This approximation, under appropriate limits, which are separately discussed, reduces either to the Zel'dovich approximation or to the adhesion one. Then the evolution of the overdensity is treated with SN equations in Zel'dovich approximation as a whole, without dividing the system into shells, obtaining results that perfectly overlap with the ones held by the shell by shell study in the Zel'dovich limit. Eventually, for a specific density profile, time dependent perturbation theory is used to refine the evolution of its shells computed in the free-particle approximation. Then it is studied the evolution of a density profile coherent with the initial conditions of the Universe which are described in literature. For this system, it is explicitly found the shell by shell exact SN solution, the SN solution in Zel'dovich approximation, and it is discussed the evolution of a mini halo placed inside it. Independently on the specific density profile considered, the exact solution prescribes that the shells of the overdensity initially expand at a slower rate than the background, then they turn around and collapse. The free-particle approximation similarly predicts that regions of the overdensity for which the density is below a critical value initially expand, then turn around and collapse; but differently, if they exist, regions whose density exceeds, at the initial time, the critical density, directly contract. In both treatments, eventually the density diverges: in the centre of symmetry of the perturbation if it is spherically symmetric, or possibly elsewhere if a test halo is added to the system. Finally, the effect on the system of a non-null cosmological constant is studied, by deriving its effect on the solution which describes a shell. For low enough cosmological constants, the evolution quantitatively resembles the one computed for the Λ=0 case.ope

Selected Topics in Gravity, Field Theory and Quantum Mechanics

Quantum field theory has achieved some extraordinary successes over the past sixty years; however, it retains a set of challenging problems. It is not yet able to describe gravity in a mathematically consistent manner. CP violation remains unexplained. Grand unified theories have been eliminated by experiment, and a viable unification model has yet to replace them. Even the highly successful quantum chromodynamics, despite significant computational achievements, struggles to provide theoretical insight into the low-energy regime of quark physics, where the nature and structure of hadrons are determined. The only proposal for resolving the fine-tuning problem, low-energy supersymmetry, has been eliminated by results from the LHC. Since mathematics is the true and proper language for quantitative physical models, we expect new mathematical constructions to provide insight into physical phenomena and fresh approaches for building physical theories