In this article, we introduce the fractional differential systems in the sense of the Weber fractional derivatives and study the asymptotic stability of these systems. We present the stability regions and then compare the stability regions of fractional differential systems with the Riemann-Liouville and Weber fractional derivatives.

The dynamical equations describing the evolution of a self-gravitating fluid can be rewritten in the form of a Schrödinger equation coupled to a Poisson equation determining the gravitational potential. This wave-mechanical representation allows an approach to cosmological gravitational instability that has numerous advantages over standard fluid-based methods. We explore the usefulness of the ...

In this article, we survey the asymptotic stability analysis of fractional differential systems with the Prabhakar fractional derivatives. We present the stability regions for these types of fractional differential systems. A brief comparison with the stability aspects of fractional differential systems in the sense of Riemann-Liouville fractional derivatives is also given. 

The spatial and time dependent solutions of the Schrödinger equation incorporating the fractional time derivative of distributed order and extending the spatial operator to noninteger dimensions are investigated. They are obtained by using the Green function approach in two situations: the free case and in the presence of a harmonic potential. The results obtained show an anomalous spreading of...

We review some recent results on eigenvalues of fractional Laplacians and fractional Schrödinger operators. We discuss, in particular, Lieb–Thirring inequalities and their generalizations, as well as semi-classical asymptotics.

This thesis is concerned with modelling the electrostatic environment in devices used for realizing Majorana zero modes (Majorana devices). A simulation model based on the SchrödingerPoisson method is developed and applied to two geometries. The first geometry is a planar semiconductor-metal heterostructure contacted to a gate electrode and with translational invariance in two directions. We sh...

Abstract. We study the Schrödinger–Poisson system on the unit sphere S2 of R3, modeling the quantum transport of charged particles confined on a sphere by an external potential. Our first results concern the Cauchy problem for this system. We prove that this problem is regularly well-posed on every Hs(S2) with s > 0, and not uniformly well-posed on L2(S2). The proof of well-posedness relies on ...

In this paper, we are concerned with seeking exact solutions for fractional differential-difference equations by an extended Riccati sub-ODE method. The fractional derivative is defined in the sense of the modified Riemann-liouville derivative. By a combination of this method and a fractional complex transformation, the iterative relations from indices n to n ± 1 are established. As for applica...