Let us consider the multiple integral $$I_{n}=\int_{-\infty }^{\infty }ds_{1}\int_{-\infty}^{s_{1}}ds_{2}\cdots \int_{-\infty }^{s_{2n-1}}ds_{2n}\;\cos {(s_{1}^{2}-s_{2}^{2})}\;\cdots \cos {(s_{2n-1}^{2}-s_{2n}^{2})}.$$
Using an indirect method (Landau-Zener formula from the theory of molecular scattering), we had shown in http://arxiv.org/abs/1201.1975 that $$I_n=\frac{2}{n!}\left(\frac{\pi}{4}\right)^{\,n}.$$ I'm very interested how to get this result by direct mathematical (preferably simple) methods, without using the Landau-Zener formula. Can anyone help?
Actually, I asked similar question an year ago Multiple Integral (American Mathematical Monthly problem 11621 and its generalization) but have not got an answer.
