the first unperturbed *odd* state.

-   Similarly, each higher even level acquires an energy corresponding to the neighbouring unperturbed odd level.

-   When $V_1R_1^2$ reaches about 2.46, enabling the narrow square well to support a second state, a new rearrangement is observed, with, again, values close to these of the unperturbed odd spectrum.

In short, the energies corresponding to the even states of the initial spectrum quickly disappear. The energies corresponding to the odd states remain, and become almost degenerate, except when a rearrangement occurs.

The degeneracy observed in Fig. \[OddEvenSq\] depends crucially on the addtional potential being of very short range. For comparison the case of a wider range $R_1=0.1$ is shown in Fig. \[OddEvenSqa\]. Though the rearrangement pattern is clearly visible, the transition is much smoother, and the almost degeneracy limited to smaller intervals of the coupling constant $V_1$, and less pronounced.

Spectrum in an asymmetric potential
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To check the interpretation of the patterns observed for the odd and even parts of the spectrum, let us break parity and consider the asymmetric double well of Fig. \[DW1DaP\]. For the sake of illustration, the centre of the spike is taken at $R_0=0.1$. The spectrum, as a function of 