The structure of semiclassical asymptotic expansions of antisymmetric solutions of the stationary Schrodinger equation

We consider the semiclassical asymptotics of eigenfunctions for the Hamiltonian of a quantum-mechanical system of N identical fermions with d degrees of freedom without spin interaction. In the one-dimensional case (d = 1), examples are known in which the ground antisymmetric state in the semiclassical limit is the product of N(N - 1)/2 two-particle wave functions. We construct a nontrivial generalization of this property for d > 1.