SDVR of the Free Particle Propagator

We now give the sinc-function based DVR of the power series Green s function. For simplicity, we restrict our attention to a one-dimensional system. The multidimensional generalization is straightforward, and will be given afterwards. Letting 

To obtain the SDVR of the free particle propagator, one takes the infinite N limit of Eq. (3.23) keeping Ax fixed, giving [97] 

With the variable transformation p = firry/Ax, the free particle propagator in Eq. (3.24) takes on the more familiar form 

The matrix element in Eq. (3.25) is noteworthy in three respects. First, it is the product of an integration weight and a finite grid spacing representation of the kernel. This product arises because DVR includes integration weights in the 

erf(2) is the error function of a complex variable [98]. The third, and perhaps most intriguing aspect of Eqs. (3.25), (3.27), and (3.28) is that they have been derived before in a completely different context by Makri [69]. In particular, she was seeking a well behaved (i.e. less oscillatory) representation of the short-time kernel for use in real time path integral Monte Carlo calculations [63-68, 70-76] The advantage gained from the matrix element in Eqs. (3.27) and (3.28) derives from the asymptotic behavior of the smoothing function [69, 81], namely 

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