A product formula for the eigenfunctions of a quartic oscillator

Abstract

We consider the Schrödinger 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 ${\psi }_k$.

18 pages. In v2 we added five references, reorganised some of the material and made some minor revisions and corrections; and in v3 we added references to work by T. T. Truong, who obtained a product formula for quartic oscillator eigenfunctions already in 1974

Cited by in corpus (1)

• On Razamat's $A_2$ and $A_3$ kernel identities