One-dimensional Schrodinger Equation Research Articles

A New Two-step Obrechkoff Method with Vanished Phase-lag and Some of its Derivatives for the Numerical Solution of Radial Schrodinger Equation and Related IVPs with Oscillating Solutions

A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

Superconvergence of Local Discontinuous Galerkin Method for One-Dimensional Linear Schrödinger Equations

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrodinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the $$L^{2}$$ norm. The order of superconvergence is $$2k+1$$ , when the polynomials of degree at most k are used. Even though the linear Schrodinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a $$(2k+1)$$ -th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of $$(k+2)$$ -th order at the Radau points. All theoretical findings are confirmed by numerical experiments.

A solution of the one-dimensional nonlinear logarithmic Schrodinger equation

A solution of the one-dimensional nonlinear logarithmic Schrodinger equation

Fermi–Pasta–Ulam recurrence and modulation instability

We give a qualitative explanation of the analog of the Fermi-Pasta-Ulam (FPU) recurrence in a one-dimensional focusing nonlinear Schrodinger equation (NLSE). That recurrence can be considered as a result of the nonlinear development of modulation instability. All known exact localized solitons-type solutions describing propagation on the background of the modulationally unstable condensate show the recurrence to the condensate state after its interaction with solitons. The condensate state locally recovers its original form with the same amplitude but a different phase after soliton leave its initial region. This is the analog of the FPU recurrence for the NLSE. Based on the integrability of the NLSE, we demonstrate that the FPU recurrence takes place not only for condensate but for more general solution in the form of the cnoidal wave. This solution is periodic in space and can be represented as a solitonic lattice. That lattice reduces to isolated soliton solution in the limit of large distance between solitons. The lattice transforms into the condensate in the opposite limit of dense soliton packing. The cnoidal wave is also modulationally unstable due to soliton overlapping. This instability at the linear stage does not provide the cnoidal wave recurrence. The recurrence happens at the nonlinear stage of the modulation instability. From the practical point of view the latter property is very important, especially for the fiber communication systems which use soliton as an information carrier.

Quark-anti-quark potential in N $$ \mathcal{N} $$ = 4 SYM

We construct a closed system of equations describing the quark--anti-quark potential at any coupling in planar N=4 supersymmetric Yang-Mills theory. It is based on the Quantum Spectral Curve method supplemented with a novel type of asymptotics. We present a high precision numerical solution reproducing the classical and one-loop string predictions very accurately. We also analytically compute the first 7 nontrivial orders of the weak coupling expansion. Moreover, we study analytically the generalized quark--anti-quark potential in the limit of large imaginary twist to all orders in perturbation theory. We demonstrate how the QSC reduces in this case to a one-dimensional Schrodinger equation. In the process we establish a link between the Q-functions and the solution of the Bethe-Salpeter equation.

Dynamics of solitons in the one-dimensional nonlinear Schrödinger equation

We investigate bright solitons in the one-dimensional Schrodinger equation in the framework of an extended variational approach. We apply the latter to the stationary ground state of the system as well as to coherent collisions between two or more solitons. Using coupled Gaussian trial wave functions, we demonstrate that the variational approach is a powerful method to calculate the soliton dynamics. This method has the advantage that it is computationally faster compared to numerically exact grid calculations. In addition, it goes far beyond the capability of analytical ground state solutions, because the variational approach provides the ability to treat excited solitons as well as dynamical interactions between different wave packets. To demonstrate the power of the variational approach, we calculate the stationary ground state of the soliton and compare it with the analytical solution showing the convergence to the exact solution. Furthermore, we extend our calculations to nonstationary solitons by investigating coherent collisions of several wave packets in both the low- and high-energy regime. Comparisons of the variational approach with numerically exact simulations on grids reveal excellent agreement in the high-energy regime while deviations can be observed for low energies.

Asymptotic integration of a certain second-order linear delay differential equation

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrodinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.

Toward a classification of quasirational solutions of the nonlinear Schrödinger equation

Based on a representation in terms of determinants of the order 2N, we attempt to classify quasirational solutions of the one-dimensional focusing nonlinear Schrodinger equation and also formulate several conjectures about the structure of the solutions. These solutions can be written as a product of a t-dependent exponential times a quotient of two N(N+1)th degree polynomials in x and t depending on 2N−2 parameters. It is remarkable that if all parameters are equal to zero in this representation, then we recover the PN breathers.

Classical or Quantum? A Computational Study of Small Ion Diffusion in II–VI Semiconductor Quantum Dots

Ion diffusion in semiconductor nanocrystals, or quantum dots (QDs), has gained recognition in recent years as a crucial process for advancing both energy storage and, more generally, the postsynthetic p-type doping chemistry of these materials. In this report, we present first an energetic analysis of group I cations (H+, Li+, and Na+) diffusion in (MX)84– QDs, with M = Zn, Cd and X = S, Se. The bound solutions to the corresponding one-dimensional nuclear Schrodinger equation were solved for these systems, relying on the discrete variable representation method. From this vantage, the quantum nature of the intercalating ion can be revealed. Evidence for the importance of including quantum effects in the treatment of these diffusion processes is presented, both with the density of energy eigenstates of the intercalating ion and from a comparison of the standard deviation in the population distribution of the intercalating ion to the lattice spacings of its host material. Results suggest that the use of clas...

Schrödinger potentials solvable in terms of the confluent Heun functions

We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrodinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice.

Schwarz waveform relaxation method for one-dimensional Schrödinger equation with general potential

In this paper, we apply the Schwarz waveform relaxation (SWR) method to the one-dimensional Schrodinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schrodinger equation with time-independent linear potential, which is robust and scalable up to 500 subdomains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time-dependent linear potential or the nonlinear potential, we use a preprocessed linear operator for the zero potential case as a preconditioner which leads to a preconditioned algorithm. This ensures high scalability. In addition, some newly constructed absorbing boundary conditions are used as the transmission conditions and compared numerically.

Gibbs measures associated to the integrals of motion of the periodic derivative nonlinear Schrödinger equation

We study the one-dimensional periodic derivative nonlinear Schrodinger equation. This is known to be a completely integrable system, in the sense that there is an infinite sequence of formal integrals of motion \({\textstyle \int }h_k\), \(k\in {\mathbb {Z}}_{+}\). In each \({\textstyle \int }h_{2k}\) the term with the highest regularity involves the Sobolev norm \(\dot{H}^{k}({\mathbb {T}})\) of the solution of the DNLS equation. We show that a functional measure on \(L^2({\mathbb {T}})\), absolutely continuous w.r.t. the Gaussian measure with covariance \(({\mathbb {I}}+(-\varDelta )^{k})^{-1}\), is associated to each integral of motion \({\textstyle \int }h_{2k}\), \(k\ge 1\).

Numerical Verification of Transition’s Energies of Excitons in Quantum Well of ZnO with the Finite Difference Method

This paper shows that the experimental results of quantum well energy transitions can be found numerically. The cases of several ZnO-ZnMgO wells are considered and their excitonic transition energies were calculated using the finite difference method. In that way, the one-dimensional Schrodinger equation has been solved by using the BLAS and LAPACK libraries. The numerical results are in good agreement with the experimental ones.

Compression of extreme ultraviolet pulse for atom with resonant structure exposed to an infrared laser field

The short attosecond (as) pulse is a basic tool for probing the ultra-fast electronic dynamics in matter. High-order harmonic generation (HHG) of atoms exposed to intense laser field is the most promising method of producing the short attosecond pulses. Therefore, the generation of ultra-short attosecond pulses through HHG has been of great interest. How to obtain the ultra-short pulse from HHG has been a hot research subject in recent years. In the present paper, we investigate the characteristic of HHG from atoms with both resonant and non-resonant structure (for short, the general atom) by using numerically solving a one-dimensional time-dependent Schrodinger equation of atom driven by two-color field (infrared (IR) laser + extreme ultraviolet (XUV)). We find that the HHG spectra from resonant atom are obviously different from those of the general atom. For a resonant atom, besides the great increase of the intensity of HHG at some energy (resonant energy + ionized energy), the intensity of HHG at the central frequency of XUV pulse is sensitive to the intensity of XUV pulse. Even the intensity of XUV pulse is weak, the enhancement of HHG spectra from resonant atom is still significant, while the general atom does not has this feature. Only the strength of the XUV pulse is much stronger than that in the case of resonant atom, the spectra of HHG near the center frequency of XUV from atom with non-resonant structure can significantly be enhanced. More importantly, adjusting the time delay of two-color laser pulse makes the width of input XUV pulse compressed obviously in the case of the resonant atom. By performing the time-frequency analysis of Morlet transform, we explain the compression of the attosecond pulse. The reason is that the relation of the input XUV pulse frequency to the resonant frequency of HHG for resonant atom makes the bandwidth of HHG in the region of the center frequency of XUV wider than that of the input attosecond pulse during the emission. Thus, we can obtain shorter pulse by superposing several orders HHG among the enhanced regions. Finally, we propose a way to compress the width of the input XUV pulse by using filter-multi-feedback method. Based on our scheme, the width of the input XUV pulse can be compressed from 200 as to 120 as, thereby offering a new method of obtaining shorter attosecond pulse in experiment.

Adiabatic Evolution Generated by a Schrödinger Operator with Discrete and Continuous Spectra

In the paper, we consider the one-dimensional nonstationary Schrodinger equation with a potential slowly depending on time. It is assumed that the corresponding stationary operator depending on time as a parameter has a finite number of negative eigenvalues and absolutely continuous spectrum filling the positive semiaxis. A solution close at some moment to an eigenfunction of the stationary operator is studied. We describe its asymptotic behavior in the case where the eigenvalues of the stationary operator move to the edge of the continuous spectrum and, having reached it, disappear one after another.

Classical trajectories as solutions of the Schrödinger equation

A method is developed for deriving solutions to a classical equation of motion with a time-dependent mass and frequency in terms of exact solutions to the one-dimensional Schrodinger equation. The method works in two directions. Analytical solutions of the classical equation of motion may be used to find the interaction potential to exactly solve the Schrodinger equation. Two simple examples have been considered that illustrate possible applications of the method.

Potential and matrix elements of the hamiltonian of internal rotation in molecules in the basis set of Mathieu functions

The advantages of the orthonormal basis set of 2π-periodic Mathieu functions compared to the trigonometric basis set in calculations of torsional states of molecules are substantiated. Explicit expressions are derived for calculating the Hamiltonian matrix elements of a one-dimensional torsional Schrodinger equation with a periodic potential of the general form in the basis set of Mathieu functions. It is shown that variation of a parameter of Mathieu functions allows the rotation potential and the structural function to be approximated with a good accuracy by a small number of series terms. The conditions for the best choice of this parameter are specified, and approximations are obtained for torsional potentials of n-butane upon rotation about the central C-C bond and of its univalent radical n-butyl C2H5C·H2 upon rotation of the C·H2 group. All algorithms are implemented in the Maple package.

Superconvergence of Local Discontinuous Galerkin methods for one-dimensional linear parabolic equations

In this paper, we study the superconvergence properties of the LDG method for the one-dimensional linear Schrodinger equation. We build a special interpolation function by constructing a correction function, and prove the numerical solution is superclose to the interpolation function in the \(L^{2}\) norm. The order of superconvergence is \(2k+1\), when the polynomials of degree at most k are used. Even though the linear Schrodinger equation involves only second order spatial derivative, it is actually a wave equation because of the coefficient i. It is not coercive and there is no control on the derivative for later time based on the initial condition of the solution itself, as for the parabolic case. In our analysis, the special correction functions and special initial conditions are required, which are the main differences from the linear parabolic equations. We also rigorously prove a \((2k+1)\)-th order superconvergence rate for the domain, cell averages, and the numerical fluxes at the nodes in the maximal and average norm. Furthermore, we prove the function value and the derivative approximation are superconvergent with a rate of \((k+2)\)-th order at the Radau points. All theoretical findings are confirmed by numerical experiments.

Effect of position-dependent effective mass on optical properties of spherical nanostructures

The effects of position-dependent effective mass on the absorption coefficients (ACs) of spherical quantum dot (QD) and quantum antidot (QAD) have been investigated. We show that the radial position-dependent effective mass Schrodinger equation can be reduced into ordinary one-dimensional Schrodinger equation with some suitable transformations. Our numerical calculations are performed using both a constant effective mass and the position-dependent effective mass. The comparative approach is used for presenting the results of both QD and QAD models. It is found that not only for the effective constant mass but also for the position-dependent effective mass, the ACs curves corresponding to QD and QAD are considerably different in shapes and behaviors. Our obtained results show that the spatial variation of the electron effective mass can not be neglected in the optical ACs of both QD and QAD nanostructures.

A New Multiscale Discontinuous Galerkin Method for the One-Dimensional Stationary Schrödinger Equation

In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schrodinger equations with open boundary conditions which have highly oscill...