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 grows, 3)
those due to divergence at very small
, and 4) those due to the appearance of spurious surface-wave
solutions in the `useful'
window.
The first of these is already familiar from the previous chapter, and
Section 6.3.3.
It places a minimum possible tension
on each state--the basis cannot produce a smaller tension
with any choice of coefficient vector
.
This tension is therefore that of the function
alone
(see (6.17)).
This is visible as the bottoming-out at
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.