Investigation of scattering processes in quantum few-body systems involving long-range interaction by the complex-rotation method

The complex-rotation method adapted to solving the multichannel scattering problem in the two-body system where the interaction potential contains the long-range Coulomb components is described. The scattering problem is reformulated as the problem of solving a nonhomogeneous Schrödinger equation in which the nonhomogeneous term involves a Coulomb potential cut off at large distances. The incident wave appearing in the nonhomogeneous term is a solution of the Schrödinger equation with longrange Coulomb interaction. This formulation is free from approximations associated with a direct cutoff of Coulomb interaction at large distances. The efficiency of this formalism is demonstrated by considering the example of solving scattering problems in the α-α and p-p systems.     

Quantum-hydrodynamic approach to the problem of electron exchange between atomic particles and nanosystems

The application of quantum-hydrodynamic methods for solving the problem of electron exchange between atomic particles and solid surfaces, and nanosystems has been examined. The derivation of a system of equations that is alternative to the nonstationary Schrödinger equation is given to describe the dynamics of electronic processes with variable charge and current densities. A comparison of results of solving the nonstationary Schrödinger equation and the quantum-hydrodynamic system of equations shows that both approaches give a good coincidence. The numerical solution to the system of quantum-hydrodynamic equations has a number of advantages, because it does not lead to oscillations at the boundary of the computational mesh and nor to the problem of exponential growth in numerical complexity for many-electron systems.     

Polarization effects and energy band diagram in AlGaN/GaN heterostructure

We report on a classical approach used to calculate energy band diagrams of AlGaN/GaN heterostructures. We were able to calculate the band diagram and carrier concentrations by this method also when the external bias was applied on the structure. The potential on the Schottky barrier side of the structure is defined more exactly than in a self-consistent solution of Poisson and Schrödinger equations. Dependence of the band profile and the carrier concentration of the two-dimensional gas on the piezoelectric charge can also be calculated by this approach.     

Anomalous suppression of the probability of resonance interaction of electrons with an rf field in asymmetric double-barrier structures on account of plasma oscillations

An analytical self-consistent solution of the nonstationary Schrödinger and Poisson equations describing the resonance interaction of electrons with a radio-frequency (rf) field has been found for asymmetric resonance-tunneling structures with high, thin barriers. It is shown that plasma oscillations limit the probability of transitions between neighboring levels and suppress anomalously strongly transitions in which the level number changes by more than 1 (by tens and hundreds of times compared with transitions between neighboring levels).     

Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations

Stochastic wave equations of Schrödinger type are widely employed in physics and have numerous potential applications in chemistry. While some accurate numerical methods exist for particular classes of stochastic differential equations they cannot generally be used for Schrödinger equations. Efficient and accurate methods for their numerical solution therefore need to be developed. Here we show that existing Runge–Kutta methods for ordinary differential equations (odes) can be modified to solve stochastic wave equations provided that appropriate changes are made to the way stepsizes are selected. The order of the resulting stochastic differential equation (sde) scheme is half the order of the ode scheme. Specifically, we show that an explicit 9th order Runge–Kutta method (with an embedded 8th order method) for odes yields an order 4.5 method for sdes which can be implemented with variable stepsizes. This method is tested by solving systems of equations originating from master equations and from the many-body Schrödinger equation.     

On- and off-shell Jost functions and their integral representations

By judicious exploitation of the transpose operator relation in conjunction with the differential equations of special functions of mathematical physics, integral representations of the on- and off-shell Jost functions are derived from the particular integrals of the inhomogeneous Schrödinger equation. Using the particular integral of the inhomogeneous Schrödinger equation, exact analytical expressions for the Coulomb and Coulomb plus Yamaguchi off-shell Jost solutions are constructed in the maximal reduced form. As a case study, the limiting behaviours and the on-shell discontinuities of the Coulomb plus Yamaguchi Jost solutions are verified numerically.     

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.     

Simulation of the interplay between stimulated emission and carrier distribution in quantum-cascade lasers

We present a model for the simulation of the effect of stimulated emission on the transport and optical properties in quantum-cascade lasers. The model is based on the self-consistent solution of the Schrödinger and Poisson equations using a one-dimensional scattering rate approach, which includes the laser rate equations. We discuss the charge redistribution, the modification of the current density, and the shift of the gain maximum for various designs of mid-infrared as well as THz quantum-cascade lasers. We found that this shift varies for the different designs, but is of similar order of magnitude as due to typical fluctuations of the layer parameters such as thicknesses and composition. In some cases, the inclusion of stimulated emission results in the appearance of negative differential conductivity, which may explain the observed instabilities of the current and light output power.     

Influence of the initial conditions on the dynamics of nonlinear systems

The nonlinear dynamics of nondissipative systems of hydrodynamic type that are subjected to small, albeit finite, perturbations in the initial conditions is considered. The behavior of these systems is determined by equations of low-energy hydrodynamics with the potential obeying the Poisson equation. The relation of the hydrodynamic system under study to systems whose dynamics is determined by the Schrödinger equation and the wave equation is discussed.     

New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

耦合广义非线性薛定谔方程的相互作用表象龙格库塔算法及其误差分析

本文通过表象变换, 将耦合广义非线性薛定谔方程 (C-GNLSE) 变换成相互作用表象中的向量方程, 再利用向量形式的4阶龙格-库塔迭代格式, 建立了一种在频域内求解C-GNLSE的同步更新迭代算法. 通过将该向量形式的相互作用表象中的4阶龙格-库塔 (V-JH-RK4IP) 算法应用于高双折射光子晶体光纤中超连续谱产生的数值模拟, 验证了算法的有效性, 通过与现有其他典型算法的比较, 表明以V-JH-RK4IP算法求解C-GNLSE具有最高的计算精度和计算效率. 关键词： 耦合广义非线性薛定谔方程(C-GNLSE) 相互作用表象 4阶龙格-库塔算法 超连续谱产生     

On Asymptotic Nonlocal Symmetry of Nonlinear Schrödinger Equations

Abstract

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schrödinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schrödinger equations. An important class of solutions of the free Schrödinger equation with improved smoothing properties is obtained.     

Self-consistent modeling ofC–Vand electronic properties of strained heterostructure modulation-doped field-effect transistors

A self-consistent solution of Schrödinger and Poisson equations is implemented in order to provide a model for the capacitance-voltage characteristics of strained modulation-doped field-effect transistors. The proposed Hamiltonian accounts for the strain caused by lattice mismatch, as well as the position dependent lattice constant and electron effective mass. The inclusion of these strain-related energy terms is shown to be paramount to achieve good agreement between theory and experimental data for theC–Vcharacteristics of pseudomorphic MODFETs at the forward bias range.     

Scattering of electrons in the GaAs/AlAs transistor structure

This paper reports on the results of a self-consistent calculation of the rates of electron scattering from surface roughnesses, acoustic phonons, and polar optical phonons in a transistor structure based on a GaAs quantum wire in an AlAs matrix at temperatures T = 77 and 300 K. The rates of electron scattering are calculated in the electric-quantum limit approximation with due regard for both the collisional broadening of the electron energy spectrum and the Pauli principle. The influence of the gate voltage on these rates is investigated. The wave function of electrons and the energy level of their quantum ground state are determined by the self-consistent solution of the Poisson and Schrödinger equations.     

Electron structure of interstitial hydrogen inα-Zr

The impurity-induced charge density in jellium is calculated by solving the Schr?dinger equation self-consistently. The resulting phase shifts have been used to estimate the value of residual resistivity for dilute Zr-H system, which comes out to be 0.50 μΘ cm/at.%. An alternative form of one-parameter-screened Coulomb potential, which is more suitable than the customary Thomas-Fermi potential, is suggested. The calculated self-energy by using new potential is found close to its value obtained by Darbyet al.     

Generalized linear Schrödinger equations and their Darboux transformations

We construct explicit Darboux transformations of arbitrary order for a class of generalized, linear Schrödinger equations. Our construction contains the well-known Darboux transformations for Schrödinger equations with position-dependent mass, Schrödinger equations coupled to a vector potential and Schrödinger equations for weighted energy.     

The Aharonov–Bohm effect in noncommutative quantum mechanics

The Aharonov–Bohm effect in noncommutative (NC) quantum mechanics is studied. First, by introducing a shift for the magnetic vector potential we give the Schrödinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then, by solving the Schrödinger equations, we obtain the Aharonov–Bohm phase on NC space and NC phase space, respectively.     

Inverse scattering method and vector higher order non-linear Schrödinger equation

A generalized inverse scattering method has been developed for arbitrary n-dimensional Lax equations. Subsequently, the method has been used to obtain N-soliton solutions of a vector higher order non-linear Schrödinger equation, proposed by us. It has been shown that under a suitable reduction, the vector higher order non-linear Schrödinger equation reduces to the higher order non-linear Schrödinger equation. An infinite number of conserved quantities have been obtained by solving a set of coupled Riccati equations. Gauge equivalence is shown between the vector higher order non-linear Schrödinger equation and the generalized Landau–Lifshitz equation and the Lax pair for the latter equation has also been constructed in terms of the spin field, establishing direct integrability of the spin system.