Separation of Variables for the Hamilton–Jacobi Equation on Complex Projective Spaces

Abstract

The additive separation of variables in the Hamilton-Jacobi equation and the multiplicative separation of variables in the Laplace-Beltrami equation are studied for the complex projective space $\mathbb{C}P^n $ considered as a Riemannian Einstein space with the standard Fubini–Study metric. The isometry group of $\mathbb{C}P^n $ is $su(n + 1)$ and its Cartan subgroup is used to generate n ignorable variables (variables not figuring in the metric tensor). A one-to-one correspondence is established between separable coordinate systems on $S^n$ and separable systems with n ignorable variables on $\mathbb{C}P^n $. The separable coordinates in $\mathbb{C}P^n $ are characterized by $2n$ integrals of motion in involution: n of them are elements of the Cartan subalgebra of $SU(n + 1)$ and the remaining n are linear combinations of the Casimir operators of $n(n + 1)$ different $su(2)$ subalgebras of $SU(n + 1)$. Each system of $2n$ integrals of motion in involution, and hence each separable system of coordinates o...