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 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.

- Deterioration of the eigenstates with
- Breakdown for a state as vanishes
- Spurious surface-wave solutions