A product formula for the eigenfunctions of a quartic oscillator
journal contributionposted on 17.03.2015 by Martin Hallnas, Edwin Langmann
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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.
This work was supported by the Göran Gustafsson Foundation (grant No. GGS 1221) and the Swedish Research Council (VR) under contract No.621-2010-3708.
- Mathematical Sciences