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A Preconditioned Iterative Solver for the Scattering Solutions of the Schr¨odinger Equation

Communications in Computational Physics 2012年第2期11卷 415-434页

作者： Hisham bin Zubair Bram Reps wim vanroose Department of Mathematical Sciences Faculty of Computer ScienceInstitute of Business AdministrationUniversity Rd.75270 KarachiPakistan Department of Mathematics and Computer Science University of AntwerpMiddelheimlaan 1B-2020 AntwerpenBelgium

The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in *** demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system,whereby each diagonal block is a high frequency Helmholtz *** on this model,we derive indefinite Helmholtz model problems with strongly varying *** employ the iterative approach for their *** 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 *** multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model *** aim of this study is to report the feasibility of this preconditioner for the model *** compare this idea with the other prevalent preconditioning ideas,and discuss its *** of numerical experiments are presented,which complement the proposed ideas,and show that this preconditioner may be used in an automatic setting.

关键词： Scattering Schrodinger equation exterior complex scaling preconditioning multigrid complex-shifted Laplacian(CSL) complex-scaled grid(CSG) quadrant-definite(QD)

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Numerical Continuation of Resonances and Bound States in Coupled Channel Schr¨odinger Equations

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Communications in Computational Physics 2012年第2期11卷 435-455页

作者： Przemysław Kłosiewicz Jan Broeckhove wim vanroose Department of Mathematics and Computer Science Universiteit AntwerpenMiddelheimlaan 1B-2020

In this contribution,we introduce numerical continuation methods and bifurcation theory,techniques which find their roots in the study of dynamical systems,to the problem of tracing the parameter dependence of bound and resonant states of the quantum mechanical Schrodinger *** extend previous work on the subject[1]to systems of coupled *** and resonant states of the Schrodinger equation can be determined through the poles of the S-matrix,a quantity that can be derived from the asymptotic form of the wave *** introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties,thus making it amenable to numerical *** allows us to automate the process of tracking bound and resonant states when parameters in the Schrodinger equation are *** have applied this approach to a number of model problems with satisfying results.

关键词： Resonances numerical continuation coupled channels Schrodinger equation

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