Abstract With the aim to solve time-dependent Schr ̈odinger equation associated a non-Hermitian Hamiltonian, we introduce unitary transformation that maps Hamiltonian time-independent PT -symmetric one. Consequently, solution of Schrödinger becomes easily deduced and evolution preserves C ( t ) -inner product, where is obtained from charge conjugation operator through time dependent transformati...

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...

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...

Abstract We prove an existence and uniqueness theorem for exact WKB solutions of general singularly perturbed linear second-order ODEs in the complex domain. These include one-dimensional time-independent Schrödinger equation. Notably, our results are valid both case generic trajectories as well closed trajectories. also explain what sense formal form a basis. As corollary proof, we establish B...

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...

With the use of a variant of the method of separation of variables, the initial value problem for the time-dependent linear Schrödinger equation is solved exactly for a large class of potential functions related to multisoliton interactions in the vector nonlinear Schrödinger equation. Completeness of states is proved for absolutely continuous initial data in L1. Copyright © 1998 Elsevier Scien...

A class of stochastic particle models for the spatially discretized time-dependent Schrödinger equation is constructed. Each particle is characterized by a complex-valued weight and a position. The particle weights change according to some deterministic rules between the jumps. The jumps are determined by the creation of offspring. The main result is that certain functionals of the particle sys...