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We study the Hartree equation with a slowly varying smooth potential, V (x) = W (hx), and with an initial condition that is ε ≤ √ h away in H 1 from a soliton. We show that up to time |log h|/ h and errors of size ε + h 2 in H 1 , 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.

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