A product formula for the eigenfunctions of a quartic oscillator

We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the typeψk(x)ψk(y)=∫Rψk(z)K(x,y,z)dz for its eigenfunctions ψkψk. The kernel function KK is given explicitly in terms of the Airy function Ai(x)Ai(x), and it is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions ψkψk.