A time-splitting pseudospectral method for the solution of the Gross-Pitaevskii equations using spherical harmonics with generalised-Laguerre basis functions

We present a method for numerically solving a Gross-Pitaevskii system of equations with a harmonic and a toroidal external potential that governs the dynamics of one- and two-component Bose-Einstein condensates. The method we develop maintains spectral accuracy by employing Fourier or spherical harmonics in the angular coordinates combined with generalised-Laguerre basis functions in the radial direction. Using an error analysis, we show that the method presented leads to more accurate results than one based on a sine transform in the radial direction when combined with a time-splitting method for integrating the equations forward in time. In contrast to a number of previous studies, no assumptions of radial or cylindrical symmetry is assumed allowing the method to be applied to 2D and 3D time-dependent simulations. This is accomplished by developing an efficient algorithm that accurately performs the generalised-Laguerre transforms of rotating Bose-Einstein condensates for different orders of the Laguerre polynomials. Using this spatial discretisation together with a second order Strang time-splitting method, we illustrate the scheme on a number of 2D and 3D computations of the ground state of a non-rotating and rotating condensate. Comparisons between previously derived theoretical results for these ground state solutions and our numerical computations show excellent agreement for these benchmark problems. The method is further applied to simulate a number of time-dependent problems including the Kelvin-Helmholtz instability in a two-component rotating condensate and the motion of quantised vortices in a 3D condensate

An algebraic method to solve the radial Schrödinger equation

AbstractWe propose a method of numerical integration of differential equations of the type x2y″+f(x)y=0 by approximating its solution with solutions of equations of the type x2y″+(ax2+bx+c)y=0. This approximation is performed by segmentary approximation on an interval. We apply the method to obtain approximate solutions of the radial Schrödinger equation on a given interval and test it for two different potentials. We conclude that our method gives a similar accuracy than the Taylor method of higher order

Solution of the Time-Dependent Schrödinger Equation for Highly Symmetric Potentials

The method of symmetry adapted wavepackets (SAWP) to solve the time-dependent Schrödinger equation for a highly symmetric potential energy surface is introduced. The angular dependence of a quantum-mechanical wavepacket is expanded in spherical harmonics where the number of close-coupled equations for the corresponding radial functions can be efficiently reduced by symmetry adaption of the rotational basis using the SAWP approach. Various techniques to generate symmetry adapted spherical harmonics (SASHs) for the point groups of highest symmetry (octahedral, icosahedral) are discussed. The standard projection operator technique involves the use of Wigner rotation matrices. Two methods to circumvent numerical instabilities occuring for large azimuthal quantum numbers are suggested. The first is based on a numerical scheme which employs Gaussian integrations yielding exact and stable results. The second is a recursive algorithm to generate higher order SASHs accurately and efficiently from lower order ones. The paper gives a complete set of ``seed functions'' generated by projection techniques which can be used to obtain SASHs for all irreducible representations of the octahedral and icosahedral point groups recursively

Computing excited states of molecules using normalizing flows

We present a new nonlinear variational framework for simultaneously computing ground and excited states of quantum systems. Our approach is based on approximating wavefunctions in the linear span of basis functions that are augmented and optimized \emph{via} composition with normalizing flows. The accuracy and efficiency of our approach are demonstrated in the calculations of a large number of vibrational states of the triatomic H$_2$S molecule as well as ground and several excited electronic states of prototypical one-electron systems including the hydrogen atom, the molecular hydrogen ion, and a carbon atom in a single-active-electron approximation. The results demonstrate significant improvements in the accuracy of energy predictions and accelerated basis-set convergence even when using normalizing flows with a small number of parameters. The present approach can be also seen as the optimization of a set of intrinsic coordinates that best capture the underlying physics within the given basis set

Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions

In this work, we address the problem of solving nonlinear general Klein–Gordon equations (nlKGEs). Different fourth- and sixth-order, stable explicit and implicit, finite difference schemes are derived. These new methods can be considered to approximate all type of Klein–Gordon equations (KGEs) including phi-four, forms I, II, and III, sine-Gordon, Liouville, damped Klein–Gordon equations, and many others. These KGEs have a great importance in engineering and theoretical physics.The higher-order methods proposed in this study allow a reduction in the number of nodes, which might also be very interesting when solving multi-dimensional KGEs. We have studied the stability and consistency of the proposed schemes when considering certain smoothness conditions of the solutions. Additionally, both the typical Dirichlet and some nonlocal integral boundary conditions have been studied. Finally, some numerical results are provided to support the theoretical aspects previously considered