Improved numerical approximation of the time kernel factor using hybrid schemes

Abstract

Using finite-difference approximation, we have developed four difference schemes which utilize the known symmetry of the time kernel factor. The schemes are combined to obtain two numerical algorithms to approximate the solution of the time kernel equation. We use the algorithms to calculate the time kernel factor of the harmonic oscillator in a confined region. Our result shows that in −1 ≤ q ≤ 1 ∩ −1 ≤ q′ ≤ 1 we were able to approximate the kernel factor of harmonic oscillator with maximum relative error of ∼ 10⁻¹¹. This means that our poorest calculation yields 11 correct significant figures.