Test of Quantum Action for Inverse Square Potential

We present a numerical study of the quantum action previously introduced as a parametrisation of Q.M. transition amplitudes. We address the questions: Is the quantum action possibly an exact parametrisation in the whole range of transition times (\(0 < T < \infty\))? Is the presence of potential terms beyond those occuring in the classical potential required? What is the error of the parametrisation estimated from the numerical fit? How about convergence and stability of the fitting method (dependence on grid points, resolution, initial conditions, internal precision etc.)? Further we compare two methods of numerical determination of the quantum action: (i) global fit of the Q.M. transition amplitudes and (ii) flow equation. As model we consider the inverse square potential, for which the Q.M. transition amplitudes are analytically known. We find that the relative error of the parametrisation starts from zero at T=0 increases to about \(10^{-3}\) at \(T=1/E_{gr}\) and then decreases to zero when \(T \to \infty\). Second, we observe stability of the quantum action under variation of the control parameters. Finally, the flow equation method works well in the regime of large \(T\) giving stable results under variation of initial data and consistent with the global fit method.