Tight 9-designs on two concentric spheres

Abstract: The main purpose of this paper is to show the nonexistence of tight Euclidean 9-designs on 2 concentric spheres in $\mathbb R^n$ if $n\geq 3.$ This in turn implies the nonexistence of minimum cubature formulas of degree 9 (in the sense of Cools and Schmid) for any spherically symmetric integrals in $\mathbb R^n$ if $n\geq 3.$