= Apparent level crossing from a finite-interval oscillator perturbation
{title2=$H'=\tfrac12m\omega^2(a^2-x^2)\mathbf1_{[-a,a]}$}
Let $\sigma^2=\hbar/(2m\omega)$ and $g(x)=(2\pi\sigma^2)^{-1/2}e^{-x^2/(2\sigma^2)}$ be the <quantum harmonic oscillator> ground-state <probability density>. Put $p(a)=\int_{-a}^ag(x)\,dx$. The first excited-state density is $x^2g(x)/\sigma^2$. <Integration by parts>, using $g'=-xg/\sigma^2$, gives the first-order difference
$$
E'_1-E'_0=-\frac{\hbar\omega}{2}[p(a)-2ag(a)].
$$
The bracket grows from zero to one, since its derivative is $2a^2g(a)/\sigma^2$. Thus the first-order predicted level gap for $H_0+\lambda H'$ is $\hbar\omega[1-\lambda\{p(a)-2ag(a)\}/2]$. For every $\lambda>2$ it becomes negative at sufficiently large $a$.
For $\lambda>1$ the full potential has a central barrier and symmetric minima at $\pm a$. The unperturbed even <ground state> samples the central barrier more strongly than the odd <excited state>, whose node reduces its shift. This explains the perturbative tendency, but the negative predicted gap is a failure diagnostic of the first-order truncation. The exact one-dimensional <ground state> remains nodeless and below the first excited state; in a high symmetric barrier the actual pair has a small positive <double-well tunneling splitting>.
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