  We consider the Schr\"odinger operator on the real line with an even quartic
potential. Our main result is a product formula of the type $\psi_k(x)\psi_k(y)
= \int_{\mathbb{R}} \psi_k(z)\mathcal{K}(x,y,z)dz$ for its eigenfunctions
$\psi_k$. The kernel function $\mathcal{K}$ is given explicitly in terms of the
Airy function $\mathrm{Ai}(x)$, and is positive for appropriate parameter
values. As an application, we obtain a particular asymptotic expansion of the
eigenfunctions $\psi_k$.
