10.5 There and back again - an exploration of the liouville  (Page 7/10)

The following tables display the error in the first 10 recovered eigenvalues. Listed are the eigenvalues numerically recovered from $q\left(t\right)$ and those recovered from its transformed counterpart, $\rho \left(x\right)$ . As expected, those recovered from $q\left(t\right)$ are accurate to over ten decimal digits. This is a consequence of the accuracy of the spectral method with sufficient Chebyshev points. The error in the eigenvalues recovered from $\rho \left(x\right)$ reflects the numerical error in our ODE solver. However, the error is tolerable. In addition, we see that the error in the transformation of the $q\left(t\right)$ defined by ${\mu }_1=9$ is less than for the other choices of ${\mu }_1$ listed in the tables.

For the majority of our research, the focus was on the potential function [link] from chapter 6 of [link] . In the next section, some results will be discussed for other $q\left(t\right)$ or $\rho \left(x\right)$ in closed form.

Corresponding $q\left(t\right)$ And $\rho \left(x\right)$ Functions in closed form

The potential function $q\left(t\right)$ given by [link] from chapter 6 of [link] , after taking the second derivative of the Wronskian, becomes