Embedded methods for the numerical solution of the Schrödinger equation

Abstract New methods for the approximate numerical integration of the one-dimensional Schrodinger equation are developed in this paper. Complete phase-lag analysis of the new methods is included. These new methods are called embedded methods because of a simple natural error control mechanism. Numerical results obtained for a one-dimensional Schrodinger equation show the validity of the developed theory.

[1]  Ben P. Sommeijer,et al.  Explicit Runge-Kutta (-Nyström) methods with reduced phase errors for computing oscillating solutions , 1987 .

[2]  T. E. Simos.  Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schro¨dinger equation , 1992 .

[3]  T. E. Simos,et al.  Numerical integration of the one-dimensional Schro¨dinger equations , 1990 .

[4]  A. C. Allison,et al.  Exponential-fitting methods for the numerical solution of the schrodinger equation , 1978 .

[5]  A. D. Raptis,et al.  A high order method for the numerical integration of the one-dimensional Schrödinger equation , 1984 .

[6]  M. M. Chawla,et al.  An explicit sixth-order method with phase-lag of order eight for y ″= f ( t , y ) , 1987 .

[7]  M. Rizea,et al.  A numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies , 1980 .

[8]  L. Brusa,et al.  A one‐step method for direct integration of structural dynamic equations , 1980 .

[9]  Majida Kobeissi,et al.  A new variable step method for the numerical integration of the one-dimensional Schro¨dinger equation , 1988 .

[10]  R. Thomas,et al.  Phase properties of high order, almostP-stable formulae , 1984 .

[11]  A. D. Raptis,et al.  On the numerical solution of the Schrödinger equation , 1981 .

[12]  T. E. Simos.  New variable-step procedure for the numerical integration of the one-dimensional Schro¨dinger equation , 1993 .

[13]  A. D. Raptis,et al.  Exponential and Bessel fitting methods for the numerical solution of the Schrödinger equation , 1987 .

[14]  T. E. Simos.  A four-step method for the numerical solution of the Schro¨dinger equation , 1990 .

[15]  H. A. Watts,et al.  Solving Nonstiff Ordinary Differential Equations—The State of the Art , 1976 .

[16]  T. E. Simos,et al.  A sixth-order exponentially fitted method for the numerical solution of the radial , 1990 .

[17]  A. D. Raptis,et al.  Exponentially-fitted solutions of the eigenvalue Shrödinger equation with automatic error control , 1983 .

[18]  T. E. Simos.  Some New Four-Step Exponential-Fitting Methods for the Numerical Solution of the Radical Schrödinger Equation , 1991 .

[19]  Veerle Fack,et al.  A finite difference approach for the calculation of perturbed oscillator energies , 1985 .

[20]  Majida Kobeissi,et al.  On testing difference equations for the diatomic eigenvalue problem , 1988 .

[21]  John M. Blatt,et al.  Practical points concerning the solution of the Schrödinger equation , 1967 .

[22]  Veerle Fack,et al.  Extended) Numerov method for computing eigenvalues of specific Schrodinger equations , 1987 .

[23]  J. W. Cooley,et al.  An improved eigenvalue corrector formula for solving the Schrödinger equation for central fields , 1961 .

[24]  Tom E. Simos.  Exponential fitted methods for the numerical integration of the Schrdinger equation , 1992 .

[25]  I. Gladwell,et al.  Damping and phase analysis for some methods for solving second‐order ordinary differential equations , 1983 .

[26]  M H Chawla,et al.  A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value , 1986 .

[27]  P. J. Van Der Houmen,et al.  Predictor-corrector methods for periodic second-order initial-value problems , 1987 .

[28]  John P. Coleman,et al.  Numerical Methods for y″ =f(x, y) via Rational Approximations for the Cosine , 1989 .

[29]  M. M. Chawla,et al.  Two-step fourth-order P-stable methods with phase-lag of order six for y ″=( t,y ) , 1986 .

[30]  J. Killingbeck.  Direct expectation value calculations , 1985 .

[31]  J. Killingbeck,et al.  Shooting methods for the Schrodinger equation , 1987 .

[32]  Tom E. Simos.  Error analysis of exponential-fitted methods for the numerical solution of the one-dimensional Schrödinger equation , 1993 .

[33]  T. E. Simos,et al.  A two-step method for the numerical solution of the radial Schrödinger equation , 1995 .

[34]  A. D. Raptis,et al.  A variable step method for the numerical integration of the one-dimensional Schrödinger equation , 1985 .

[35]  A. Messiah.  Quantum Mechanics , 1961 .