An accurate finite difference method for the numerical solution of the Schrِdinger equation

An accurate finite difference approach for computing eigenvalues of Schrödinger equations is developed in this paper. We investigate two cases: (i) the specific case in which the potential V(x) is an even function with respect to x. It is assumed, also, that the wave functions tend to zero for x → ±∞. We investigate the well-known potential of the onedimensional anharmonic oscillator, the symmetric double-well potential, the Razavy potential and the doubly anharmonic oscillator potential. (ii) The general case for positive and negative eigenvalues and for the well-known cases of the Morse potential and Woods-Saxon or optical potential. Numerical and theoretical results show that this new approach is more efficient than previously derived methods.