A new methodology for the construction of numerical methods for the approximate solution of the Schr?dinger equation

In the present paper we introduce a new methodology for the development of numerical methods for the numerical solution of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications. T. E. Simos is Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts. Corresponding Member of the European Academy of Sciences and European Academy of Arts, Sciences and Humanities.     

High order multistep methods with improved phase-lag characteristics for the integration of the Schr?dinger equation

In this work we introduce a new family of 12-step linear multistep methods for the integration of the Schr?dinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications. T. E. Simos is a highly cited researcher, active member of the European Academy of Sciences and Arts. Corresponding member of the European Academy of Sciences, corresponding member of European Academy of Arts, Sciences and Humanities.     

A new methodology for the development of numerical methods for the numerical solution of the Schr?dinger equation

In the present paper we introduce a new methodology for the construction of numerical methods for the approximate solution of the one-dimensional Schr?dinger equation. The new methodology is based on the requirement of vanishing the phase-lag and its derivatives. The efficiency of the new methodology is proved via error analysis and numerical applications.     

Closed Newton–Cotes trigonometrically-fitted formulae of high order for the numerical integration of the Schrödinger equation

In this paper we investigate the connection between (i) closed Newton–Cotes formulae, (ii) trigonometrically-fitted differential methods, (iii) symplectic integrators and (iv) efficient solution of the Schrödinger equation. In the last decades several one step symplectic integrators have been produced based on symplectic geometry, (see the relevant literature and the references here). However, the study of multistep symplectic integrators is very poor. In this paper we investigate the closed Newton–Cotes formulae and we write them as symplectic multilayer structures. We also develop trigonometrically-fitted symplectic methods which are based on the closed Newton–Cotes formulae. We apply the symplectic schemes to the well known radial Schrödinger equation in order to investigate the efficiency of the proposed method to these type of problems.