Using an appropriate scaling group for the 3-D Schrödinger–Poisson equation and the equivalence between the Schrödinger formalism and the Wigner representation of quantum mechanics it is proved that, when time goes to infinity, the limit of the rescaled self-consistent potential can be identified as the Coulomb potential. As a consequence, Schrödinger–Poisson and Wigner–Poisson systems are asym...

Ehrenfest, Born-Oppenheimer, Langevin and Smoluchowski dynamics are shown to be accurate approximations of time-independent Schrödinger observables for a molecular system avoiding caustics, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger...

In this paper we have solved analytically the Schrödinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. Based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square ...

We consider the real stationary two-dimensional Schrödinger equation. With the aid of any its particular solution we construct a Vekua equation possessing the following special property. The real parts of its solutions are solutions of the original Schrödinger equation and the imaginary parts are solutions of an associated Schrödinger equation with a potential having the form of a potential obt...

We analyze the solutions of the Schrödinger equation with the low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a non-trivial energy in the high frequencies, which do not homogenize – the high frequency component of the wave field remains random and...

This paper deals with the asymptotic behaviour of the solutions of linear initial boundary value problems with constant coefficients on the half-line and on finite intervals. We assume that the boundary data are periodic in time and we investigate whether the solution becomes time-periodic after sufficiently long time. Using the Fokas’ transformation method, we show that for the linear Schrödin...

Interaction and correlation effects in quantum dots play a fundamental role in defining both their equilibrium and transport properties. Numerical methods are commonly employed to study such systems. In this paper we investigate the numerical calculation of quantum transport of electrons in spherical centered defect InGaAs/AlGaAs quantum dot (SCDQD). The simulation is based on the imaginary time...

1 Schrödinger Equation and Time Development Operators 1 1.1 Schrödinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Time-dependent kets and operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Solutions to Schrödinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Time developme...

We consider a nonlinear Schrödinger equation in R3 with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinearexcited state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrödinger equation approac...

We consider the 3–dimensional Schrödinger operator with slowly decaying potential whose radial derivatives are short-range. The long-time asymptotics for solution of the corresponding non-stationary equation is established. In this case, the standard WKB-correction should be replaced by the solution of certain evolution equation. Consider the Schrödinger operator H = −∆+ V, x ∈ R (1) In this pa...