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In this paper, we investigate the connection between closed Newton-Cotes formulae, trigonometrically-fitted methods, symplectic integrators and efficient integration of the Schr¨odinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton-Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the onedimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schr¨odinger equation in order to investigate the efficiency of the proposed method to these type of problems.

Many simulation algorithms (chemical reaction systems, differential systems arising from the modeling of transient behavior in the process industries and etc.) contain the numerical solution of systems of differential equations. For the efficient solution of the above mentioned problems, linear multistep methods or Runge-Kutta technique are used. For the simulation of chemical procedures the radial Schrodinger equation is used frequently. In the present paper we will study a symmetric two-step Obrechkoff method, in which we will use of technique of VSDPL (vanished some of derivatives of phase-lag), for the numerical integration of the one-dimensional Schrodinger equation. We will show superiority of new method in stability, accuracy and efficiency. So we present a stability analysis and an error analysis based on the radial Schrodinger equation. Also we will apply the new proposed method to the resonance problem of the radial Schrodinger equation.