The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear...The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system,whereby each diagonal block is a high frequency Helmholtz problem.Based on this model,we derive indefinite Helmholtz model problems with strongly varying wavenumbers.We employ the iterative approach for their solution.In 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 components.This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems.The aim of this study is to report the feasibility of this preconditioner for the model problems.We compare this idea with the other prevalent preconditioning ideas,and discuss its merits.Results of numerical experiments are presented,which complement the proposed ideas,and show that this preconditioner may be used in an automatic setting.展开更多
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 a...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 equation.We extend previous work on the subject[1]to systems of coupled equations.Bound 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 function.We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties,thus making it amenable to numerical continuation.This allows us to automate the process of tracking bound and resonant states when parameters in the Schrodinger equation are varied.We have applied this approach to a number of model problems with satisfying results.展开更多
基金funded partially by Fonds voor Wetenschappelijk Onderzoek(FWO Bel-gium)projects G.0174.08 and 1.5.145.10,by the University of Antwerp,Belgium,and by the Institute of Business Administration,Karachi,Pakistan.We wish to thank the sponsors sincerely for their support.
文摘The Schrodinger equation defines the dynamics of quantum particles which has been an area of unabated interest in physics.We demonstrate how simple transformations of the Schrodinger equation leads to a coupled linear system,whereby each diagonal block is a high frequency Helmholtz problem.Based on this model,we derive indefinite Helmholtz model problems with strongly varying wavenumbers.We employ the iterative approach for their solution.In 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 components.This multigrid preconditioner is used in conjunction with suitable Krylov-subspace methods for solving the indefinite Helmholtz model problems.The aim of this study is to report the feasibility of this preconditioner for the model problems.We compare this idea with the other prevalent preconditioning ideas,and discuss its merits.Results of numerical experiments are presented,which complement the proposed ideas,and show that this preconditioner may be used in an automatic setting.
文摘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 equation.We extend previous work on the subject[1]to systems of coupled equations.Bound 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 function.We introduce a regularization procedure that essentially transforms the S-matrix into its inverse and improves its smoothness properties,thus making it amenable to numerical continuation.This allows us to automate the process of tracking bound and resonant states when parameters in the Schrodinger equation are varied.We have applied this approach to a number of model problems with satisfying results.
