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We consider random Schrödinger equations on$R$^{$d$}for$d$≽ 3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$ . The space and time variables scale as $ x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)} $ . We prove that, in the limit λ → 0, the expectation of the Wigner distribution of $\psi_t$ converges weakly to the solution of a heat equation in the space variable$x$for arbitrary$L$^{2}initial data.

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Let$U$be an open subset of a complex locally convex space$E$, and$H(U)$the space of holomorphic functions from$U$to$C$. If the dual$E′$of$E$is nuclear with respect to the topology generated by the absolutely convex compact subsets of$E$, then it is shown that$H(U)$endowed with the compact open topology is a nuclear space. In particular, if$E$is the strong dual of a Fréchet nuclear space, then$H(U)$is a Fréchet nuclear space.