imagine a different Coulomb system for each $n$, with $\alpha\propto n$, hence a Bohr radius independent of $n$, and a wave function at the origin $|\phi(0)|^2\propto n^{-1}$ instead of $n^{-3}$ in the usual case. Then, in this situation, $\delta E\propto n^{-1}$ for the Coulomb system, and $\delta \epsilon\propto n^{1/2}$ for the harmonic oscillator.

Rearrangement and level ordering {#se:order}
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In the above examples, there is an interesting superposition of potentials with different level-ordering properties. A square well potential, if deep enough to support many bound states, has the ordering [@Landau] $$\label{eq:sqorder}
1S<2P<3D<2S<\ldots~.$$ We are adopting here the same notation as in atomic physics is adopted, i.e., 2P is the first P-state, 3D the first D-state, etc. The Coulomb potential, on the other hand, exhibits the well-known degeneracy $$\label{eq:cborder}
1S<2S=2P<3S=3P=3D<\ldots~,$$ while for the harmonic-oscillator case, $$\label{eq:hoorder}
1S<2P<2S=3D<\ldots~,$$ with equal spacing.

The pattern of 1S, 2S and 2P levels for Coulomb (left) or harmonic oscillator (right) supplemented by a short-range square well of increasing strength is given in Fig. \[fig:crossing\].

In the Coulomb case, the degeneracy is broken at small $\lambda$ as $E(2S)<E(2P)$ since the 2P wave function vanishes at $r=0$. The 2S drops when the 1S state falls into the region 