An accurate finite difference method for the numerical solution of the Schrödinger equation

An accurate finite difference approach for computing eigenvalues of Schrödinger equations is developed in this paper. We investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x. It is assumed, also, that the wave functions tend to zero for x → ±∞. We investigate the well-known potential of the onedimensional anharmonic oscillator, the symmetric double-well potential, the Razavy potential and the doubly anharmonic oscillator potential. (ii) The general case for positive and negative eigenvalues and for the well-known cases of the Morse potential and Woods-Saxon or optical potential. Numerical and theoretical results show that this new approach is more efficient than previously derived methods.     

Homotopy perturbation method for coupled Schrödinger–KdV equation

In this paper, numerical analysis of the coupled Schrödinger–KdV equation is studied by using the Homotopy Perturbation Method (HPM). The available analytical solutions of the coupled Schrödinger–KdV equation obtained by multiple traveling wave method are compared with HPM to examine the accuracy of the method. The numerical results validate the convergence and accuracy of the Homotopy Perturbation Method for the analyzed coupled Schrödinger–KdV equation.     

The Laguerre pseudospectral method for the radial Schrödinger equation

By transforming dependent and independent variables, radial Schrödinger equation is converted into a form resembling the Laguerre differential equation. Therefore, energy eigenvalues and wavefunctions of M-dimensional radial Schrödinger equation with a wide range of isotropic potentials are obtained numerically by using Laguerre pseudospectral methods. Comparison with the results from literature shows that the method is highly competitive.     

A finite-difference method for the numerical solution of the Schrödinger equation

A new approach, which is based on a new property of phase-lag for computing eigenvalues of Schrödinger equations with potentials, is developed in this paper. We investigate two cases: (i) The specific case in which the potential V(x) is an even function with respect to x. It is assumed, also, that the wave functions tend to zero for x → ±∞. (ii) The general case of the Morse potential and of the Woods-Saxon or optical potential. Numerical and theoretical results show that this new approach is more efficient compared to previously derived methods.     

A numerical method for solving the time fractional Schrödinger equation

In this article, we proposed a new numerical method to obtain the approximation solution for the time-fractional Schrödinger equation based on reproducing kernel theory and collocation method. In order to overcome the weak singularity of typical solutions, we apply the integral operator to both sides of differential equation and yield a integral equation. We divided the solution of this kind equation into two parts: imaginary part and real part, and then derived the approximate solutions of the two parts in the form of series with easily computable terms in the reproducing kernel space. New bases of reproducing kernel spaces are constructed and the existence of approximate solution is proved. Numerical examples are given to show the accuracy and effectiveness of our approach.     

Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.     

Existence of solutions to the nonlinear Schrödinger equation on locally finite graphs

Archiv der Mathematik - Let $$G=(V, E)$$ be a locally finite connected graph and $$\Delta $$ be the usual graph Laplacian operator. According to Lin and Yang (Rev. Mat. Complut., 2022), using...     

High-order linear compact conservative method for the nonlinear Schrödinger equation coupled with the nonlinear Klein–Gordon equation

In this paper, we design a linear-compact conservative numerical scheme which preserves the original conservative properties to solve the Klein–Gordon–Schrödinger equation. The proposed scheme is based on using the finite difference method. The scheme is three-level and linear-implicit. Priori estimate and the convergence of the finite difference approximate solutions are discussed by the discrete energy method. Numerical results demonstrate that the present scheme is conservative, efficient and of high accuracy.     

A nonlinear nonstandard finite difference scheme for the linear time-dependent Schrödinger equation

We state and study the various limiting forms and their associated mathematical properties of a nonlinear finite difference scheme for the linear time-dependent Schrödinger partial differential equation (PDE). A formal solution to the full equation is given.     

Adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices

The Ramanujan Journal - In this article, we obtain the adjoint difference equation for the Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices,...     

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.     

Fractional Schrödinger equation; solvability and connection with classical Schrödinger equation

We consider the Dirichlet boundary problem for semilinear fractional Schrödinger equation with subcritical nonlinear term. Local and global in time solvability and regularity properties of solutions are discussed. But our main task is to describe the connections of the fractional equation with the classical nonlinear Schrödinger equation, including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent α approaches 1.     

Non-Nehari manifold method for periodic discrete superlinear Schrödinger equation

We consider the nonlinear difference equations of the form Lu=f(n, u), n∈Z, where L is a Jacobi operator given by(Lu)(n)=a(n)u(n+1)+a(n-1)u(n-1)+b(n)u(n) for n∈Z, {a(n)} and {b(n)} are real valued N-periodic sequences, and f(n, t) is superlinear on t. Inspired by previous work of Pankov [Discrete Contin. Dyn. Syst., 19, 419-430(2007)] and Szulkin and Weth [J. Funct. Anal., 257, 3802-3822(2009)], we develop a non-Nehari manifold method to find ground state solutions of Nehari-Pankov type under weaker conditions on f. Unlike the Nehari manifold method, the main idea of our approach lies on finding a minimizing Cerami sequence for the energy functional outside the Nehari-Pankov manifold by using the diagonal method.     

Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity

Groundstates of the stationary nonlinear Schrödinger equation $-\Delta u +V u =K u^{p-1}$ , are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in ${L^2(\mathbb R^N)}Groundstates of the stationary nonlinear Schr?dinger equation

-Du +V u = K up-1-\Delta u +V u =K u^{p-1}      15. On the statistical derivation of the Schrödinger equation      Yu. L. Klimontovich 《Theoretical and Mathematical Physics》1993,97(1):1111-1125 The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of simple systems—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a point of a continuous medium is described by the classical electron radiusr e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr e and the corresponding time interval e =r e /c play the role of hidden parameters in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient = e –1 , the presence of which reflects loss of information on smoothing over the volume of a point of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the collision integrals being expressed in terms of the correlation functions of these fluctuations.In memory of Dmitrii Nikolaevich ZubarevMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 3–26, October, 1993.      16. Self-similar solutions for the Schrödinger map equation      Pierre Germain  Jalal Shatah  Chongchun Zeng 《Mathematische Zeitschrift》2010,264(3):697-707 We study in this article the equivariant Schrödinger map equation in dimension 2, from the Euclidean plane to the sphere. A family of self-similar solutions is constructed; this provides an example of regularity breakdown for the Schrödinger map. These solutions do not have finite energy, and hence do not fit into the usual framework for solutions. For data of infinite energy but small in some norm, we build up associated global solutions.      17. Exact solutions of the nonstationary Schrödinger equation      E. P. Velicheva  A. A. Suz'ko 《Theoretical and Mathematical Physics》1999,121(3):1700-1700       18. CP methods for the Schrödinger equation revisited      《Journal of Computational and Applied Mathematics》1998,88(2):289-314       19. An asymptotic expression of the Schrödinger equation      Zhaosheng Feng  David Y. Gao 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,84(2):363-375 The problem of solving the time–independent Schr?dinger equation for the motion of an electron of mass μ and charge –e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equations to the study of the Schr?dinger equation under certain parametric conditions, and obtain an asymptotic formula. The work has been presented at the International Conference on Quantum Computation and Quantum Technology, Texas A&M University, College Station, Texas, November 13-16, 2005. The author would like to thank the organizer Professor Goong Chen for his generous support. This work is also partly supported by UTPA Faculty Research Council Grant 119100.      20. Existence time for the semilinear Schrödinger equation      Wei Mingjun 《高校应用数学学报(英文版)》2003,18(1):30-34 Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L p ? L q estimate and the energy estimate.

On the statistical derivation of the Schrödinger equation

The transition from reversible microscopic operator equations to irreversible equations for a deterministic density matrix is considered for examples of simple systems—the hydrogen atom or a free electron in an electromagnetic field. As a result of the transition, a system of a particle and field oscillators is replaced by a continuous medium. The Schrödinger equation for the deterministic wave function also describes the evolution of a continuum but without allowance for dissipative terms. In this sense, there is an analogy between the Schrödinger equation in quantum mechanics and Euler's equation in hydrodynamics. The smallest size of a point of a continuous medium is described by the classical electron radiusr _e . It also determines the effective Thomson cross section for scattering of photons by free electrons. The lengthr _e and the corresponding time interval _e =r _e /c play the role of hidden parameters in quantum mechanics. Two methods of calculating the effective Thomson cross section in terms of the extinction coefficient are considered. The first of them is based on the equation of motion of a free electron in a field with allowance for radiative friction. This equation leads to well-known difficulties. Moreover, the velocity fluctuations calculated on its basis lead to a contradiction with the second law of thermodynamics. The second method is based on the introduction of a constant friction coefficient = _e ^–1 , the presence of which reflects loss of information on smoothing over the volume of a point of the continuous medium. Such a method of calculation leads to the same expression for the effective cross section but makes it possible to avoid the difficulties with the second law of thermodynamics. In the derivation of quantum kinetic equations, the physically infinitesimally small scales are determined by the Compton length _C and the corresponding time interval. The introduction of these scales makes it possible to separate and eliminate small-scale fluctuations, the collision integrals being expressed in terms of the correlation functions of these fluctuations.In memory of Dmitrii Nikolaevich ZubarevMoscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 97, No. 1, pp. 3–26, October, 1993.     

Self-similar solutions for the Schrödinger map equation

We study in this article the equivariant Schrödinger map equation in dimension 2, from the Euclidean plane to the sphere. A family of self-similar solutions is constructed; this provides an example of regularity breakdown for the Schrödinger map. These solutions do not have finite energy, and hence do not fit into the usual framework for solutions. For data of infinite energy but small in some norm, we build up associated global solutions.     

An asymptotic expression of the Schrödinger equation

The problem of solving the time–independent Schr?dinger equation for the motion of an electron of mass μ and charge –e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equations to the study of the Schr?dinger equation under certain parametric conditions, and obtain an asymptotic formula. The work has been presented at the International Conference on Quantum Computation and Quantum Technology, Texas A&M University, College Station, Texas, November 13-16, 2005. The author would like to thank the organizer Professor Goong Chen for his generous support. This work is also partly supported by UTPA Faculty Research Council Grant 119100.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.