An Exact Absorbing Boundary Condition for the Schrödinger Equation with Sinusoidal Potentials at Infinity

In this paper we study numerical issues related to the Schrödinger equation with sinusoidal potentials at infinity. An exact absorbing boundary condition in a form of Dirichlet-to-Neumann mapping is derived. This boundary condition is based on an analytical expression of the logarithmic derivative of the Floquet solution to Mathieu's equation, which is completely new to the author's knowledge. The implementation of this exact boundary condition is discussed, and a fast evaluation method is used to reduce the computation burden arising from the involved half-order derivative operator. Some numerical tests are given to show the performance of the proposed absorbing boundary conditions.     

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.     

A Time-Space Adaptive Method for the Schrödinger Equation

In this paper, we present a discretization of the time-dependent Schrödinger equation based on a Magnus-Lanczos time integrator and high-order Gauss-Lobatto finite elements in space. A truncated Galerkin orthogonality is used to obtain duality-based a posteriori error estimates that address the temporal and the spatial error separately. Based on this theory, a space-time adaptive solver for the Schrödinger equation is devised. An efficient matrix-free implementation of the differential operator, suited for spectral elements, is used to enable computations for realistic configurations. We demonstrate the performance of the algorithm for the example of matter-field interaction.     

Explicit Symplectic Methods for the Nonlinear Schrödinger Equation

By performing a particular spatial discretization to the nonlinear Schrödinger equation (NLSE), we obtain a non-integrable Hamiltonian system which can be decomposed into three integrable parts (L-L-N splitting). We integrate each part by calculating its phase flow, and develop explicit symplectic integrators of different orders for the original Hamiltonian by composing the phase flows. A 2nd-order reversible constructed symplectic scheme is employed to simulate solitons motion and invariants behavior of the NLSE. The simulation results are compared with a 3rd-order non-symplectic implicit Runge-Kutta method, and the convergence of the formal energy of this symplectic integrator is also verified. The numerical results indicate that the explicit symplectic scheme obtained via L-L-N splitting is an effective numerical tool for solving the NLSE.     

A Preconditioned Iterative Solver for the Scattering Solutions of the Schrödinger Equation

The Schrödinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics. We demonstrate how simple transformations of the Schrödinger equation leads to a coupled linear system, whereby each diagonal block is a high frequency Helmholtz problem. Based on this model, we derive indefinite Helmholtz model problems with strongly varying wavenumbers. We employ the iterative approach for their solution. In particular, we develop a preconditioner that has its spectrum restricted to a quadrant (of the complex plane) thereby making it easily invertible by multigrid methods with standard components. This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems. The aim of this study is to report the feasibility of this preconditioner for the model problems. We compare this idea with the other prevalent preconditioning ideas, and discuss its merits. Results of numerical experiments are presented, which complement the proposed ideas, and show that this preconditioner may be used in an automatic setting.     

An Optimized Perfectly Matched Layer for the Schrödinger Equation

We derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.     

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations

In this review article we discuss different techniques to solve numerically the time-dependent Schrödinger equation on unbounded domains. We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works. We conclude with several numerical examples from different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.     

Efficient Grid Treatment of the Time Dependent Schrödinger Equation for Laser-Driven Molecular Dynamics

We present an efficient method to solve the time dependent Schrödinger equation for modeling the dynamics of diatomic molecules irradiated by intense ultrashort laser pulse without Born-Oppenheimer approximation. By introducing a variable prolate spheroidal coordinates and discrete variable representations of the Hamiltonian, we can accurately and efficiently simulate the motion of both electronic and molecular dynamics. The accuracy and convergence of this method are tested by simulating the molecular structure, photon ionization and high harmonic generation of $H_2^+$.     

Stochastic Multi-Symplectic Integrator for Stochastic Nonlinear Schrödinger Equation

In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations, and develop a stochastic multi-symplectic method for numerically solving a kind of stochastic nonlinear Schrödinger equations. It is shown that the stochastic multi-symplectic method preserves the multi-symplectic structure, the discrete charge conservation law, and deduces the recurrence relation of the discrete energy. Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.     

Exact Artificial Boundary Condition for the Poisson Equation in the Simulation of the 2D Schrödinger-Poisson System

We study the computation of ground states and time dependent solutions of the Schrödinger-Poisson system (SPS) on a bounded domain in 2D (i.e. in two space dimensions). On a disc-shaped domain, we derive exact artificial boundary conditions for the Poisson potential based on truncated Fourier series expansion in θ, and propose a second order finite difference scheme to solve the $r$-variable ODEs of the Fourier coefficients. The Poisson potential can be solved within $\mathcal{O}$($M NlogN$) arithmetic operations where $M,N$ are the number of grid points in $r$-direction and the Fourier bases. Combined with the Poisson solver, a backward Euler and a semi-implicit/leap-frog method are proposed to compute the ground state and dynamics respectively. Numerical results are shown to confirm the accuracy and efficiency. Also we make it clear that backward Euler sine pseudospectral (BESP) method in [33] can not be applied to 2D SPS simulation.     

The Hyperbolic Schrödinger Equation and the Quantum Lattice Boltzmann Approximation

The quantum lattice Boltzmann (qlB) algorithm solves the 1D Dirac equations and has been used to solve approximately the classical (i.e., non-relativistic)Schrödinger equation. We point out that the qlB method actually approximates thehyperbolic version of the non-relativistic Schrödinger equation, whose solution is thusobtained at the price of an additional small error. Such an error is of order of $(ω_ctau)^{−1},$ where $ω_c:=frac{mc^2}{h}$ is the Compton frequency, $ħ$ being the reduced Planck constant, $m$ the rest mass of the electrons, $c$ the speed of light, and $tau$ a chosen reference time (i.e.,1 s), and hence it vanishes in the non-relativistic limit $c → +∞.$ This asymptotic result comes from a singular perturbation process which does not require any boundarylayer and, consequently, the approximation holds uniformly, which fact is relevant inview of numerical approximations. We also discuss this occurrence more generally, forsome classes of linear singularly perturbed partial differential equations.     

Semi-Eulerian and High Order Gaussian Beam Methods for the Schrödinger Equation in the Semiclassical Regime

A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions instead of solving the dynamic ray tracing equation. The difference lies in that, we solve the ray tracing equations to determine the centers of the beams and then compute quantities of interests only around these centers. This yields effectively a local level set implementation, and the beam summation can be carried out on the initial physical space instead of the phase plane. As a consequence, it reduces the computational cost and also avoids the delicate issue of beam summation around the caustics in the Eulerian Gaussian beam method. Moreover, the semi-Eulerian Gaussian beam method can be easily generalized to higher order Gaussian beam methods, which is the topic of the second part of this paper. Several numerical examples are provided to verify the accuracy and efficiency of both the first order and higher order semi-Eulerian methods.     

Excitonic Eigenstates of Disordered Semiconductor Quantum Wires: Adaptive Wavelet Computation of Eigenvalues for the Electron-Hole Schrödinger Equation

A novel adaptive approach to compute the eigenenergies and eigenfunctions of the two-particle (electron-hole) Schrödinger equation including Coulomb attraction is presented. As an example, we analyze the energetically lowest exciton state of a thin one-dimensional semiconductor quantum wire in the presence of disorder which arises from the non-smooth interface between the wire and surrounding material. The eigenvalues of the corresponding Schrödinger equation, i.e., the one-dimensional exciton Wannier equation with disorder, correspond to the energies of excitons in the quantum wire. The wavefunctions, in turn, provide information on the optical properties of the wire. We reformulate the problem of two interacting particles that both can move in one dimension as a stationary eigenvalue problem with two spacial dimensions in an appropriate weak form whose bilinear form is arranged to be symmetric, continuous, and coercive. The disorder of the wire is modelled by adding a potential in the Hamiltonian which is generated by normally distributed random numbers. The numerical solution of this problem is based on adaptive wavelets. Our scheme allows for a convergence proof of the resulting scheme together with complexity estimates. Numerical examples demonstrate the behavior of the smallest eigenvalue, the ground state energies of the exciton, together with the eigenstates depending on the strength and spatial correlation of disorder.     

Analysis and Efficient Solution of Stationary Schrödinger Equation Governing Electronic States of Quantum Dots and Rings in Magnetic Field

In this work the one-band effective Hamiltonian governing the electronic states of a quantum dot/ring in a homogenous magnetic field is used to derive a pair/quadruple of nonlinear eigenvalue problems corresponding to different spin orientations and in case of rotational symmetry additionally to quantum number ±ℓ. We show, that each of those pair/quadruple of nonlinear problems allows for the min-max characterization of its eigenvalues under certain conditions, which are satisfied for our examples and the common InAs/GaAs heterojunction. Exploiting the minmax property we devise efficient iterative projection methods simultaneously handling the pair/quadruple of nonlinear problems and thereby saving up to 40% of the computational time as compared to the nonlinear Arnoldi method applied to each of the problems separately.     

Computing the Ground and First Excited States of the Fractional Schrödinger Equation in an Infinite Potential Well

In this paper, we numerically study the ground and first excited states of the fractional Schrödinger equation in an infinite potential well. Due to the nonlocality of the fractional Laplacian, it is challenging to find the eigenvalues and eigenfunctions of the fractional Schrödinger equation analytically. We first introduce a normalized fractional gradient flow and then discretize it by a quadrature rule method in space and the semi-implicit Euler method in time. Our numerical results suggest that the eigenfunctions of the fractional Schrödinger equation in an infinite potential well differ from those of the standard (non-fractional) Schrödinger equation. We find that the strong nonlocal interactions represented by the fractional Laplacian can lead to a large scattering of particles inside of the potential well. Compared to the ground states, the scattering of particles in the first excited states is larger. Furthermore, boundary layers emerge in the ground states and additionally inner layers exist in the first excited states of the fractional nonlinear Schrödinger equation. Our simulated eigenvalues are consistent with the lower and upper bound estimates in the literature.     

Multiscale Computations for the Maxwell–Schrödinger System in Heterogeneous Nanostructures

In this paper, we study the multiscale computations for the Maxwell–Schrödinger system with rapidly oscillating coefficients under the dipole approximation that describes light-matter interaction in heterogeneous nanostructures. The multiscale asymptotic method and an associated numerical algorithm for the system arepresented. We propose an alternating Crank–Nicolson finite element method for solving the homogenized Maxwell–Schödinger system and prove the existence of solutionsto the discrete system. Numerical examples are given to validate the efficiency and accuracy of the algorithm.     

Solutions of the 1D Coupled Nonlinear Schrödinger Equations by the CIP-BS Method

In this paper, we present solutions for the one-dimensional coupled nonlinear Schrödinger (CNLS) equations by the Constrained Interpolation Profile-Basis Set (CIP-BS) method. This method uses a simple polynomial basis set, by which physical quantities are approximated with their values and derivatives associated with grid points. Nonlinear operations on functions are carried out in the framework of differential algebra. Then, by introducing scalar products and requiring the residue to be orthogonal to the basis, the linear and nonlinear partial differential equations are reduced to ordinary differential equations for values and spatial derivatives. The method gives stable, less diffusive, and accurate results for the CNLS equations.     

Review of Feynman's Path Integral in Quantum Statistics: From the Molecular Schrödinger Equation to Kleinert's Variational Perturbation Theory

Feynman's path integral reformulates the quantum Schrödinger differential equation to be an integral equation. It has been being widely used to compute internuclear quantum-statistical effects on many-body molecular systems. In this Review, the molecular Schrödinger equation will first be introduced, together with the Born-Oppenheimer approximation that decouples electronic and internuclear motions. Some effective semiclassical potentials, e.g., centroid potential, which are all formulated in terms of Feynman's path integral, will be discussed and compared. These semiclassical potentials can be used to directly calculate the quantum canonical partition function without individual Schrödinger's energy eigenvalues. As a result, path integrations are conventionally performed with Monte Carlo and molecular dynamics sampling techniques. To complement these techniques, we will examine how Kleinert's variational perturbation (KP) theory can provide a complete theoretical foundation for developing non-sampling/non-stochastic methods to systematically calculate centroid potential. To enable the powerful KP theory to be practical for many-body molecular systems, we have proposed a new path-integral method: automated integration-free path-integral (AIF-PI) method. Due to the integration-free and computationally inexpensive characteristics of our AIF-PI method, we have used it to perform ab initio path-integral calculations of kinetic isotope effects on proton-transfer and RNA-related phosphoryl-transfer chemical reactions. The computational procedure of using our AIF-PI method, along with the features of our new centroid path-integral theory at the minimum of the absolute-zero energy (AMAZE), are also highlighted in this review.     

Efficient Splitting Methods Based on Modified Potentials: Numerical Integration of Linear Parabolic Problems and Imaginary Time Propagation of the Schrödinger Equation

We present a new family of fourth-order splitting methods with positive coefficients especially tailored for the time integration of linear parabolic problems and,in particular, for the time dependent Schrödinger equation, both in real and imaginarytime. They are based on the use of a double commutator and a modified processor, andare more efficient than other widely used schemes found in the literature. Moreover,for certain potentials, they achieve order six. Several examples in one, two and threedimensions clearly illustrate the computational advantages of the new schemes.     

Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

In this paper, we propose a wavelet collocation splitting (WCS) method, and a Fourier pseudospectral splitting (FPSS) method as comparison, for solving one-dimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics. The two methods can preserve the intrinsic properties of original problems as much as possible. The splitting technique increases the computational efficiency. Meanwhile, the error estimation and some conservative properties are investigated. It is proved to preserve the charge conservation exactly. The global energy and momentum conservation laws can be preserved under several conditions. Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.