By Andre Martinez, Vania Sordoni

The authors build an summary pseudodifferential calculus with operator-valued image, compatible for the therapy of Coulomb-type interactions, and so they use it on the research of the quantum evolution of molecules within the Born-Oppenheimer approximation, relating to the digital Hamiltonian admitting an area hole in its spectrum. particularly, they express that the molecular evolution may be decreased to the only of a approach of delicate semiclassical operators, the logo of which are computed explicitly. furthermore, they learn the propagation of yes wave packets as much as very long time values of Ehrenfest order

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**Example text**

However, its symbol is certainly uniquely determined in the relevant region of the phase space where ϕ(t) ˜ := We−itP/h ϕ0 lives (that is, on F S(ϕ(t)) ˜ in the sense of the previous chapter, and for t ∈ [0, TΩ (ϕ0 ))). Therefore, as long as we deal with h-admissible operators (that is, with operators that do not move the Frequency Set), or even with twisted h-admissible operators (that become standard h-admissible operators once conjugated with W or ZL ) it is enough, for computing the symbol A in this region, to start by performing formal computations on the operators themselves (instead of immediately using the twisted symbolic calculus, that appears to be a little bit too heavy at the beginning).

In Chapter 10, we give a way of computing easily the expansion of A up to any power of h. As an example, we compute explicitly its ﬁrst three terms (that is, up to O(h4 )). ˜ Proof – 1) Setting ϕ := e−itP /h ϕ0 , we have f (P˜ )ϕ = ϕ, and thus ih∂t Πg ϕ = Πg P˜ f (P˜ )ϕ = Π2g P˜ f (P˜ )ϕ. 1 tells us that [Πg , P˜ ]f (P˜ ) = O(h∞ ). 5), ih∂t Πg ϕ = Πg P˜ Πg f (P˜ )ϕ + O(h∞ ϕ ) = P˜ (1) Πg ϕ + O(h∞ ϕ0 ), uniformly with respect to h and t. This equation can be re-written as, ˜ (1) /h ih∂t (eitP Πg ϕ) = O(h∞ ϕ0 ), and thus, integrating from 0 to t, we obtain, ˜ (1) /h Πg ϕ = e−itP Πg ϕ0 + O(|t|h∞ ϕ0 ), uniformly with respect to h, t and ϕ0 .

Then, denoting by B 4 2 −1 symbol [( Re aj − z)( Re aj − z)ϕj + ψj ] , the standard pseudodiﬀerential calculus with operator-valued symbols shows that, ˜j (z) = O(| Im z|−N0 ) B for some N0 ≥ 1, and, ˜j (z)Bj (z) = 1 + hRj (z), B α rj (z) = where Rj (z) is a h-admissible operator with symbol rj (z) verifying ∂x,ξ 2n −Nα,j O(| Im z| ), for all α ∈ ZZ + , and for some Nα,j ≥ 1. 9), we obtain, Uj χj uj (z) = Cj (z)v + hCj (z)Uj ϕj uj (z) + O(h∞ | Im z|−N1 ) v , (1) (1) (2) (2) where Cj (z), Cj (z) are two h-admissible operators, uniformly bounded by some negative power of | Im z|, and N1 is some positive number.