= Contact shifts of two spin-one-half fermions in a harmonic trap
For two identical <fermions> of mass $m$ in the same one-dimensional <quantum harmonic oscillator>, use <center of mass> $X=(x_1+x_2)/2$ and separation $r=x_1-x_2$. The corresponding oscillator masses are $2m$ and <reduced mass> $m/2$, with the same frequency $\omega$. If $N,n$ are their occupation numbers, the unperturbed energy is $\hbar\omega(N+n+1)$. The <fermionic exchange symmetry> requires even $n$ with a <spin-one-half singlet state> and odd $n$ with a <spin-one-half triplet state>.
For a <delta potential> $\lambda\delta(r)$ the first-order shift is $\lambda|\varphi_n^{\mathrm{rel}}(0)|^2$. Every odd relative <wavefunction> vanishes at contact, so these states remain exact unaffected eigenstates of the spatial problem. For the even relative ground state, the <oscillator length> is $\sqrt2\ell$, where $\ell=\sqrt{\hbar/(m\omega)}$, and the shift is $\lambda/(\sqrt{2\pi}\ell)$, independent of $N$. This separates interaction-sensitive singlets from contact-insensitive triplets.
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