HALLNAS, M. and LANGMANN, E., 2015. A product formula for the eigenfunctions of a quartic oscillator. Journal of Mathematical Analysis and Applications, 426 (2), pp. 1012-1025.
We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the type. ψk(x)ψk(y)=∫Rψk(z)K(x,y,z)dz for its eigenfunctions ψk. The kernel function K is given explicitly in terms of the Airy function Ai(x), and it is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions ψk.