Sources of error in the method

There are four distinct causes of errors in the scaling method: 1) those due to the basis set limitations, 2) those due to deterioration of the eigenstates as $\vert\delta _\mu \vert$ grows, 3) those due to divergence at very small $\vert\delta _\mu \vert$, and 4) those due to the appearance of spurious surface-wave solutions in the `useful' $\delta$ window.

The first of these is already familiar from the previous chapter, and Section 6.3.3. It places a minimum possible tension $\epsilon_\mu$ on each state--the basis cannot produce a smaller tension with any choice of coefficient vector ${\mathbf x}$. This tension is therefore that of the function $\varepsilon _\mu{({\mathbf s})}$ alone (see (6.17)). This is visible as the bottoming-out at $t\sim10^{-11}$ in Fig. 6.9.

Here and in the future we use tension as the main measure of the error of a found eigenstate, since is gives the 2-norm of the amount by which the boundary conditions fail to be obeyed. I will now discuss the three remaining types of error.