  We show that the k-double Schur functions defined by the authors, and the
quantum double Schubert polynomials studied by Kirillov and Maeno and by
Ciocan-Fontanine and Fulton, can be obtained from each other by an explicit
rational substitution. The main new ingredient is an explicit computation of
Kostant's solution to the Toda lattice in terms of equivariant Schubert
classes.
