We study the Hartree
equation with a slowly varying smooth potential, V (x) = W(hx), and with an initial
condition that is 𝜀 ≤ away in H1 from a soliton. We show that up to time
|logh|∕h and errors of size 𝜀 + h2 in H1, the solution is a soliton evolving
according to the classical dynamics of a natural effective Hamiltonian. This
result is based on methods of Holmer and Zworski, who prove a similar
theorem for the Gross–Pitaevskii equation, and on spectral estimates for the
linearized Hartree operator recently obtained by Lenzmann. We also provide
an extension of the result of Holmer and Zworski to more general initial
conditions.
away in 