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# Schrödinger Equation (schrödinger + equation)

Distribution by Scientific Domains

Kinds of Schrödinger Equation

 non-linear schrödinger equation nonlinear schrödinger equation time-dependent schrödinger equation

## Selected Abstracts

### Concentration on curves for nonlinear Schrödinger Equations

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 1 2007
Manuel Del Pino
We consider the problem where p > 1, , > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let , be a closed curve, nondegenerate geodesic relative to the weighted arc length ,,V,, where , = (p + 1)/(p , 1) , 1/2. We prove the existence of a solution u, concentrating along the whole of ,, exponentially small in , at any positive distance from it, provided that , is small and away from certain critical numbers. In particular, this establishes the validity of a conjecture raised in 3 in the two-dimensional case. © 2006 Wiley Periodicals, Inc. [source]

### Introductory quantum physics courses using a LabVIEW multimedia module

COMPUTER APPLICATIONS IN ENGINEERING EDUCATION, Issue 2 2007
Ismael Orquín
Abstract We present the development of a LabVIEW multimedia module for introductory Quantum Physics courses and our experience in the use of this application as an educational tool in learning methodologies. The program solves the time-dependent Schrödinger equation (TDSE) for arbitrary potentials. We describe the numerical method used for solving this equation, as well as some mathematical tools employed to reduce the calculation time and to obtain more accurate results. As an illustration, we present the evolution of a wave packet for three different potentials: the repulsive barrier potential, the repulsive step potential, and the harmonic oscillator. This application has been successfully integrated in the learning strategies of the course Quantum Physics for Engineering at the Polytechnic University of Valencia, Spain. © 2007 Wiley Periodicals, Inc. Comput Appl Eng Educ. 15: 124,133, 2007; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20100 [source]

### Big Consequences of Small Changes (Non-locality and non-linearity of Hartree-Fock equations)

CONTRIBUTIONS TO PLASMA PHYSICS, Issue 7-8 2009
M.Ya. Amusia
Abstract It is demonstrated that non-locality and non-linearity of Hartree-Fock equations dramatically affect the properties of their solutions that essentially differ from solutions of Schrödinger equation with a local potential. Namely, it acquires extra zeroes, has different coordinate asymptotic, violates so-called gauge-invariance, has different scattering phases at zero energy, has in some cases several solutions with the same set of quantum numbers, usually equivalent expressions of current and Green's functions became non-equivalent. These features result in a number of consequences for probabilities of some physical processes, leading e. g. to extra width of atomic Giant resonances and enhance considerably the ionization probability of inner atomic electrons by a strong field (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Exactly solvable effective mass Schrödinger equation with coulomb-like potential

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 15 2010
C. Pacheco-García
Abstract Exactly solvable Schrödinger equation (SE) with a position-dependent mass distribution allowing Morse-like eigenvalues is presented. For this, the position-dependent mass Schrödinger equation is transformed into a standard SE, with constant mass, by means of the point canonical transformation scheme. In that method, the choice of potential for the position-dependent mass Schrödinger equation allows us to obtain the transformation that should be used to find the exactly solvable SE. As a useful application of the proposal, the equivalent of the Witten superpotential is chosen to be constant to find the position-dependent mass distribution and the exactly solvable potential V(m(x)) allowing Morse-type energy spectra. This V(m(x)) is shown to have a Coulomb potential structure and can be useful in the study of the electronic properties of materials in which the carrier effective mass depends on the position. Moreover, the worked example, the approach is general and can be applied in the search of new potentials suitable on the study of quantum chemical systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [source]

### The rotation-vibration spectrum for Scarf II potential

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 13 2010
Wen-Chao Qiang
Abstract The arbitrary l -state solutions of the Schrödinger equation with Scarf II potential are presented by taking a new approximate scheme to the centrifugal term. This is realized by expanding the variable around the minimum equilibrium point of the potential. The wave functions can be expressed by hypergeometric functions. It is shown that the calculated energy levels are in good agreement with accurate numerical ones. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010 [source]

### Isospectral potentials for the Schrödinger equation with a position-dependent mass: Free-particle potential model

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 14 2009
J. Morales
Abstract In this work, a method to obtain exactly solvable potentials for the one-dimensional Schrödinger equation with a position-dependent mass (SEPDM) and a procedure to determine their isospectral potential partners are presented. In the first case, the problem of exactly solvable SEPDM is worked by means of the point canonical transformation method which permit to convert the SEPDM problem into a standard Schrödinger-like equation with a position-independent mass (SLEPIM). In the second case, the procedure to obtain the partner isospectral potentials that fulfill with the SEPDM involves the Darboux transform applied to the SLEPIM. As example of the usefulness of the proposals, it is considered the special case of the free particle potential model as former potential of the SEPDM which leads to different exactly solvable potentials, and to their isospectral partners, depending on the choice of the position-dependent mass distribution. The proposals are general and can be used in the search of those potentials leading to exactly solvable SEPDM and their isospectral partners, which could be useful in the modeling of quantum chemical properties in condensed matter applications. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

### The correlation contracted Schrödinger equation: An accurate solution of the G -particle-hole hypervirial

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 14 2009
D. R. Alcoba
Abstract The equation obtained by mapping the matrix representation of the Schrödinger equation with the 2nd-order correlation transition matrix elements into the 2-body space is the so called correlation contracted Schrödinger equation (CCSE) (Alcoba, Phys Rev A 2002, 65, 032519). As shown by Alcoba (Phys Rev A 2002, 65, 032519) the solution of the CCSE coincides with that of the Schrödinger equation. Here the attention is focused in the vanishing hypervirial of the correlation operator (GHV), which can be identified with the anti-Hermitian part of the CCSE. A comparative analysis of the GHV and the anti-Hermitian part of the contracted Schrödinger equation (ACSE) indicates that the former is a stronger stationarity condition than the latter. By applying a Heisenberg-like unitary transformation to the G -particle-hole operator (Valdemoro et al., Phys Rev A 2000, 61, 032507), a good approximation of the expectation value of this operator as well as of the GHV is obtained. The method is illustrated for the case of the Beryllium isoelectronic series as well as for the Li2 and BeH2 molecules. The correlation energies obtained are within 98.80,100.09% of the full-configuration interaction ones. The convergence of these calculations was faster when using the GHV than with the ACSE. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

### Laser control of photodissociation process in diatomic molecule

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 14 2009
Abstract In this work, we first aim to realize the complete laser-induced photodissociation of the OH molecule, and then intend to control the wavepacket generated on the continuum state, i.e., to achieve the laser control of the above-threshold dissociation (ATD) spectrum. To numerically solve the Schrödinger equation, we adopt the split operator method (SOM), which conserves the norm of the state vector, and can treat both discrete and continuum states simultaneously and correctly. This photodissociation process induced by the multiphoton absorption involves the ATD spectrum due to the continuum-continuum transition by the intense electric field. First, we investigate the detailed mechanism of the complete photodissociation with the one-color laser pulse by changing the laser parameters. Then, we investigate the control of the ATD spectrum by using the two-color laser field, where we focus on the role of the relative phase and position of two laser pulses. To analyze the population of both discrete and continuum states involved in the resultant wavepacket, we show the effective method by means of the quasicontinuum state on the Morse potential obtained by numerically diagonalizing the Fourier grid Hamiltonian (FGH). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

### Inverse problems in quantum chemistry

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 11 2009
Jacek Karwowski
Abstract Inverse problems constitute a branch of applied mathematics with well-developed methodology and formalism. A broad family of tasks met in theoretical physics, in civil and mechanical engineering, as well as in various branches of medical and biological sciences has been formulated as specific implementations of the general theory of inverse problems. In this article, it is pointed out that a number of approaches met in quantum chemistry can (and should) be classified as inverse problems. Consequently, the methodology used in these approaches may be enriched by applying ideas and theorems developed within the general field of inverse problems. Several examples, including the RKR method for the construction of potential energy curves, determining parameter values in semiempirical methods, and finding external potentials for which the pertinent Schrödinger equation is exactly solvable, are discussed in detail. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009 [source]

### The theory of currents through small bridge molecules

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 10 2007
B. L. Burrows
Abstract A model to treat the theory of currents through small bridge molecules connected to leads is constructed using the time-dependent Schrödinger equation and the wide-band approximation to treat the leads. It is shown that the behaviour of the current through the bridge may be summarized by considering three time periods: a transient period, a quasi-steady -state period and a decaying period. The results obtained are compared with previous work and in particular it is shown that, under reasonable assumptions, they are in accord with the more conventional time-independent scattering theory approaches in the steady-state period. Illustrative calculations are presented for both chain and ring bridge molecules. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source]

### Quantum wave packet dynamics on multidimensional adaptive grids: Applications of the moving boundary truncation method

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 7 2007
Lucas R. Pettey
Abstract Recently, we reported a novel method for integrating the time dependent Schrödinger equation which used hydrodynamic quantum trajectories to adapt the boundaries of a fixed spatial grid. The moving boundary truncation (MBT) method significantly reduced the number of grid points needed to perform accurate calculations while maintaining stability during the time propagation. In this work, the method is extended to multidimensional examples. The application of MBT to scattering on 2D and 3D potential energy surfaces shows a greater decrease in the number of grid points needed compared with full fixed grids while maintaining excellent accuracy and stability, even for very long propagation times. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source]

### Application of the asymptotic iteration method to the exponential cosine screened Coulomb potential

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2007
O. Bayrak
Abstract We present the iterative solutions of the radial Schrödinger equation for the exponential cosine screened Coulomb (ECSC) potential for any n and l quantum states by applying the asymptotic iteration method (AIM). We show that it is possible to obtain the solution as accurate as the other methods without any perturbation. Furthermore, there are no tedious mathematical difficulties and restrictions on finding the energy eigenvalues for any n and l quantum numbers. Our results are in excellent agreement with the ones published in the literature. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source]

### Exact analytical solutions to the Kratzer potential by the asymptotic iteration method

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 3 2007
O. Bayrak
Abstract For any n and l values, we present a simple exact analytical solution of the radial Schrödinger equation for the Kratzer potential within the framework of the asymptotic iteration method (AIM). The exact bound-state energy eigenvalues (Enl) and corresponding eigenfunctions (Rnl) are calculated for various values of n and l quantum numbers for CO, NO, O2, and I2 diatomic molecules. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007 [source]

### Computation of the eigenvalues of the one-dimensional Schrödinger equation by symplectic methods

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 4 2006
Z. Kalogiratou
Abstract The computation of high-state eigenvalues of the one-dimensional time-independent Schrödinger equation is considered by symplectic integrators. The Schrödinger equation is first transformed into a Hamiltonian canonical equation. Yoshida-type symplectic integrators are used as well as symplectic integrators based on the Magnus expansion. Numerical results are obtained for a wide range of eigenstates of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator, and the Morse potential. The eigenvalues found by the symplectic methods are compared with the eigenvalues produced by Numerov-type methods. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006 [source]

### Convergence enhancement in the iterative solution of the second-order contracted Schrödinger equation

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2005
D. R. Alcoba
Abstract The iterative solution of the contracted Schrödinger equation is coupled with a second-order reduced density matrix purification procedure, which corrects its N- and S-representability defects, and with a regulating convergence device. An analysis of the effects of these new implementations is reported. The method is applied to the calculation of the potential energy curves of the BeH2 and Li2 molecules. The results compare very closely with those of the full configurations interaction. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source]

### Continuum eigen-functions of 1-D time-independent Schrödinger equation solved by symplectic algorithm

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 1 2005
Yue-Ying Qi
Abstract We present the symplectic algorithm for solving the continuum eigen-functions of 1-dimensional (1D) time-independent Schrödinger equation, which keeps Wronskian conservation of linearly independent continuum eigen-functions. This symplectic algorithm has been applied to the computation of the linearly independent continuum eigen-functions of both 1D soft-coulomb potential and Pöshl,Teller short-range potential as well as the radial continuum eigen-functions of hydrogen atom. The numerical results using the symplectic algorithm are in agreement with the existing theories. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005 [source]

### Designer polynomials, discrete variable representations, and the Schrödinger equation

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 4-5 2002
Charles A. Weatherford
Abstract The general procedure for constructing a set of orthonormal polynomials is given for an arbitrary positive definite weight function, w(x), in the interval [a, b]. The Lanczos method is used to generate the three-term recursion relation, which is then used to produce the polynomial coefficients. A discrete variable representation (DVR) is constructed from Gaussian nodes and weights that result from the three-term recursion relation. These are termed "designer polynomials" and the associated "designer DVRs." It will be shown by construction that every such set of "synthetic polynomials" carries an associated DVR. The term "designer" derives from the fact that the interval [a, b] and the weight function w(x) are arbitrary (except that w(x) must be positive definite on [a, b] and must have continuous derivatives except at a finite number of isolated discontinuities) and may be adapted to the physical problem of interest. The difficulties of applying a DVR to a "bare" Coulomb problem will be illustrated on a "toy" model in one dimension (1-D hydrogen atom). A solution for the 1-D Coulomb problem will be given, thereby motivating the need for designer DVRs. In doing so, a new set of polynomials is defined with a weight function w(x) = |x|kexp(,,|x|), (such that k = ,1, 0, +1, +2, ,) between the symmetrical limits [,,, +,]. These are called "synthetic Cartesian exponential polynomials (SCEP)." These polynomials are then used in a spectral and pseudospectral (DVR) representation to solve the 1-D hydrogen atom problem. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source]

### Time asymmetry, nonexponential decay, and complex eigenvalues in the theory and computation of resonance states

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 2 2002
Cleanthes A. Nicolaides
Abstract Stationary-state quantum mechanics presents no difficulties in defining and computing discrete excited states because they obey the rules established in the properties of Hilbert space. However, when this idealization has to be abandoned to formulate a theory of excited states dissipating into a continuous spectrum, the problem acquires additional interest in many fields of physics. In this article, the theory of resonances in the continuous spectrum is formulated as a problem of decaying states, whose treatment can entail time-dependent as well as energy-dependent theories. The author focuses on certain formal and computational issues and discusses their application to polyelectronic atomic states. It is argued that crucial to the theory is the understanding and computation of a multiparticle localized wavepacket, ,0, at t = 0, having a real energy E0. Assuming this as the origin, without memory of the excitation process, the author discusses aspects of time-dependent dynamics, for t , 0 as well as for t , ,, and the possible significance of nonexponential decay in the understanding of timeasymmetry. Also discussed is the origin of the complex eigenvalue Schrödinger equation (CESE) satisfied by resonance states and the state-specific methodology for its solution. The complex eigenvalue drives the decay exponentially, with a rate ,, to a good approximation. It is connected to E0 via analytic continuation of the complex self-energy function, A(z), (z is complex), into the second Riemann sheet, or, via the imposition of outgoing wave boundary conditions on the stationary state Schrödinger equation satisfied by the Fano standing wave superposition in the vicinity of E0. If the nondecay amplitude, G(t), is evaluated by inserting the unit operator I = ,dE|E>, then the resulting spectral function is real, g(E) =|<,0|E>|2, and does not differentiate between positive and negative times. The introduction of time asymmetry, which is associated with irreversibility, is achieved by starting from < ,0|,(t)e,iHt|,0 >, where ,(t) is the step function at the discontinuity point t = 0. In this case, the spectral function is complex. Within the range of validity of exponential decay, the complex spectral function is the same as the coefficient of ,0 in the theory of the CESE. A calculation of G(t) using the simple pole approximation and the constraints that t > 0 and E > 0 results in a nonexponential decay (NED) correction for t , 1/, that is different than when a real g(E) is used, representing the contribution of both "in" and "out" states. Earlier formal and computational work has shown that resonance states close to threshold are good candidates for NED to acquire nonnegligible magnitude. In this context, a pump-probe laser experiment in atomic physics is proposed, using as a paradigm the He, 1s2p24P shape resonance. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source]

### Linear augmented Slater-type orbital method for free standing clusters

JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 8 2009
K. S. Kang
Abstract We have developed a Scalable Linear Augmented Slater-Type Orbital (LASTO) method for electronic-structure calculations on free-standing atomic clusters. As with other linear methods we solve the Schrödinger equation using a mixed basis set consisting of numerical functions inside atom-centered spheres and matched onto tail functions outside. The tail functions are Slater-type orbitals, which are localized, exponentially decaying functions. To solve the Poisson equation between spheres, we use a finite difference method replacing the rapidly varying charge density inside the spheres with a smoothed density with the same multipole moments. We use multigrid techniques on the mesh, which yields the Coulomb potential on the spheres and in turn defines the potential inside via a Dirichlet problem. To solve the linear eigen-problem, we use ScaLAPACK, a well-developed package to solve large eigensystems with dense matrices. We have tested the method on small clusters of palladium. © 2008 Wiley Periodicals, Inc. J Comput Chem, 2009 [source]

### Electron correlation: The many-body problem at the heart of chemistry

JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 8 2007
David P. Tew
Abstract The physical interactions among electrons and nuclei, responsible for the chemistry of atoms and molecules, is well described by quantum mechanics and chemistry is therefore fully described by the solutions of the Schrödinger equation. In all but the simplest systems we must be content with approximate solutions, the principal difficulty being the treatment of the correlation between the motions of the many electrons, arising from their mutual repulsion. This article aims to provide a clear understanding of the physical concept of electron correlation and the modern methods used for its approximation. Using helium as a simple case study and beginning with an uncorrelated orbital picture of electronic motion, we first introduce Fermi correlation, arising from the symmetry requirements of the exact wave function, and then consider the Coulomb correlation arising from the mutual Coulomb repulsion between the electrons. Finally, we briefly discuss the general treatment of electron correlation in modern electronic-structure theory, focussing on the Hartree-Fock and coupled-cluster methods and addressing static and dynamical Coulomb correlation. © 2007 Wiley Periodicals, Inc. J Comput Chem 28: 1307,1320, 2007 [source]

### Registering the Amica electronic structure code in the Extensible Computational Chemistry Environment

JOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 3 2005
Robert J. Gdanitz
Abstract We describe the integration and use of the Amica software package ("Atoms & Molecules In Chemical Accuracy") within the Extensible Computational Chemistry Environment (Ecce). Amica is capable of accurately solving the electronic Schrödinger equation of small atoms and molecules using terms that are linear in the interelectronic distances, r12, on multireference level of theory, but it requires expert knowledge to configure and execute its algorithms. Ecce is a comprehensive suite of tools that support the computational chemistry research processes of computation setup, execution, and analysis through a convenient graphical user interface. Additionally, Ecce was architected with mechanisms to integrate alternative electronic structure codes. The successful integration of Amica within Ecce validates the architecture of the latter and brings the high-accuracy capabilities of Amica to a wider audience. © 2004 Wiley Periodicals, Inc. J Comput Chem 26: 214,225, 2005 [source]

### Students' levels of explanations, models, and misconceptions in basic quantum chemistry: A phenomenographic study

JOURNAL OF RESEARCH IN SCIENCE TEACHING, Issue 5 2009
Christina Stefani
We investigated students' knowledge constructions of basic quantum chemistry concepts, namely atomic orbitals, the Schrödinger equation, molecular orbitals, hybridization, and chemical bonding. Ausubel's theory of meaningful learning provided the theoretical framework and phenomenography the method of analysis. The semi-structured interview with 19 second-year chemistry students supplied the data. We identified four levels of explanations in the students' answers. In addition, the scientific knowledge claims reflected three main levels of models. By combining levels of explanations with levels of models, we derived four categories. Two of the categories are shades of variation in the rote-learning part of a continuum, while the other two categories are in the meaningful-learning part. All students possessed alternative conceptions some of which occurred within certain categories, while others spanned more categories. The insistence on the deterministic models of the atom, the misinterpretation of models, and the poor understanding of the current quantum concepts are main problems in the learning of the basic quantum chemistry concepts. © 2009 Wiley Periodicals, Inc. J Res Sci Teach 46: 520,536, 2009 [source]

### Non-self-adjoint boundary-value problem with discontinuous density function

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 11 2010
Abstract We determine spectrum and principal functions of the non-self-adjoint differential operator corresponding to 1-D non-self-adjoint Schrödinger equation with discontinuous density function, provide some sufficient conditions guaranteeing finiteness of eigenvalues and spectral singularities, and introduce the convergence properties of principal functions. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Exact solutions for a perturbed nonlinear Schrödinger equation by using Bäcklund transformations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2009
Hassan A. Zedan
Abstract In this paper, the method of deriving the Bäcklund transformation from the Riccati form of inverse method is presented for the perturbed nonlinear Schrödinger equation (PNSE). Consequently, the exact solutions for the PNSE can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions that are solutions of the associated AKNS, that is, a linear eigenvalues problem in the form of a system of partial differential equation. Moreover, we construct a new soliton solution from the old one and its wave function. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Exact solutions of space,time dependent non-linear Schrödinger equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2004
Hang-yu Ruan
Abstract Using a general symmetry approach we establish transformations between different non-linear space,time dependent evolution equations of Schrödinger type and their respective solutions. As a special case we study the transformation of the standard non-linear Schrödinger equation (NLS)-equation to a NLS-equation with a dispersion coefficient which decreases exponentially with increasing distance along the fiber. By this transformation we construct from well known solutions of the standard NLS-equation some new exact solutions of the NLS-equation with dispersion. Copyright 2004 John Wiley & Sons, Ltd. [source]

### The Schrödinger equation and a multidimensional inverse scattering transform

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002
Swanhild Bernstein
Abstract The Schrödinger equation is one of the most important equations in mathematics, physics and also engineering. We outline some connections between transformations of non-linear equations, the Schrödinger equation and the inverse scattering transform. After some remarks on generalizations into higher dimensions we present a multidimensional ,¯ method based on Clifford analysis. To explain the method we consider the formal solution of the inverse scattering problem for the n -dimensional time-dependent Schrödinger equations given by A.I. Nachman and M.J. Ablowitz. Copyright © 2002 John Wiley & Sons, Ltd. [source]

### Inverse scattering for the non-linear Schrödinger equation: Reconstruction of the potential and the non-linearity

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2001
Ricardo Weder
In this paper we consider the inverse scattering problem for the non-linear Schrödinger equation on the line \def\dr{{\rm d}}$$i{\partial\over\partial t}u(t,x)=-{\dr^2\over\dr x^2}u(t,x)+V_0(x)u(t,x)+\sum_{j=1}^{\infty}V_j(x)|u|^{2(j_0+j)}u(t,x)$$\nopagenumbers\end We prove, under appropriate conditions, that the small-amplitude limit of the scattering operator determines uniquely Vj, j=0,1,, . Our proof gives also a method for the reconstruction of the Vj, j=0,1,, . Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Asymptotic analysis of solutions of a radial Schrödinger equation with oscillating potential

MATHEMATISCHE NACHRICHTEN, Issue 15 2006
Sigrun Bodine
Abstract We are interested in the asymptotic behavior of solutions of a Schrödinger-type equation with oscillating potential which was studied by A. Its. Here we use a different technique, based on Levinson's Fundamental Lemma, to analyze the asymptotic behavior, and our approach leads to a complete asymptotic representation of the solutions. We also discuss formal simplifications for differential equations with what might be called "regular/irregular singular points with periodic coefficients". (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### A dual-reciprocity boundary element solution of a generalized nonlinear Schrödinger equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004
Whye-Teong Ang
Abstract A time-stepping dual-reciprocity boundary element method is presented for the numerical solution of an initial-boundary value problem governed by a generalized non-linear Schrödinger equation. To test the method, two specific problems with known exact solutions are solved. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20, 2004. [source]

### On the finite-differences schemes for the numerical solution of two dimensional Schrödinger equation

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2002
Murat Suba
Abstract In this study three different finite-differences schemes are presented for numerical solution of two-dimensional Schrödinger equation. The finite difference schemes developed for this purpose are based on the (1, 5) fully explicit scheme, and the (5, 5) Noye-Hayman fully implicit technique, and the (3, 3) Peaceman and Rachford alternating direction implicit (ADI) formula. These schemes are second order accurate. The results of numerical experiments are presented, and CPU times needed for this problem are reported. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 752,758, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10029. [source]