PROPERTIES OF THE OPTIMUM DISTRIBUTED APPROXIMATING FUNCTION CLASS PROPAGATOR FOR DISCRETIZED AND CONTINUOUS WAVE PACKET PROPAGATIONS

A new, more concise derivation of the continuous and discretized distributed approximating function (CDAF and DDAF) class free propagators and a detailed explication of their properties are presented. The DAF class propagators are characterized by three factors: namely, a real Gaussian, a unimodulator oscillatory factor, and a 'shape polynomial' of degree M (with M even) which has complex coefficients. For the continuous version of the theory, these factors are solely a function of the time step-tau and take the forms exp[-m(x'- x)2/4h-tauBAR], exp[im(x'- x)2/4h-tauBAR], and g(M)(x - x\sigma(0)=(h-tauBAR/m)1/2,tau) respectively, where sigma(0) = (h-tauBAR/m)1/2 is the width of the Gaussian envelope of the CDAF. The discrete version of the theory requires slightly different forms for these factors, because the choice of sigma(0) depends also on the grid spacing. The relationship sigma(0) = (h-tauBAR/m)1/2, which gives the optimum choice of the Gaussian envelope of the CDAF for minimizing its spread in time-tau, is derived. The relationship between the Gaussian width sigma(0) and the degree of the shape polynomial is given, and it is shown that the specification of the time step-tau is sufficient to fix all other parameters. The time step-tau is determined by the characteristics of the propagator algorithm being used.