By Andre Martinez, Vania Sordoni
The authors build an summary pseudodifferential calculus with operator-valued image, compatible for the remedy of Coulomb-type interactions, and so they use it on the learn of the quantum evolution of molecules within the Born-Oppenheimer approximation, when it comes to the digital Hamiltonian admitting a neighborhood hole in its spectrum. specifically, they express that the molecular evolution could be lowered to the single of a process of gentle semiclassical operators, the emblem of that are computed explicitly. additionally, they research the propagation of yes wave packets as much as very long time values of Ehrenfest order
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Extra info for Twisted pseudodifferential calculus and application to the quantum evolution of molecules
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Therefore, if we choose N such that 2mN −m ≥ µM +mN , that is, N ≥ +µM /m, 6. 2 tell us that h−(M +1) [g(P˜ ), Π(M ) ] is a twisted h-admissible operator, and the result follows. • We set, ˜ g := Π(g) + Π(g)∗ − 1 (g(P˜ ))Π(g)∗ + Π(g)g(P˜ )) + (1 − g(P˜ ))Π ˜ 0 (1 − g(P˜ )). 28) ˜g − Π ˜0 Π L(L2 (IRn ;H)) ˜2 − Π ˜g + Π g L(L2 (IRn ;H)) = O(h). 29) ˜ g = P˜ P˜ Π ˜ 0 (1 − g(P˜ )) Π(g) + (1 − g(P˜ )) Π(g) + Π ≥ 0, + O(h∞ ). 30) ˜ g] P˜ [f (P˜ ), Π L(L2 (IRn ;H)) = O(h∞ ). 32) ˜2 − Π ˜ g )f (P˜ ) = O(h∞ ).
1, we obtain, Uj ψj e−itP/h ϕ0 = ψj f (P˜j )Uj χj e−itP/h ϕ0 + O(h∞ ), with P˜j = Uj χj P˜ Uj−1 χj . 2, ρ(ω)Uj ψj e−itP/h ϕ0 = O(h∞ ). 7) F S(Uj e−itP/h ϕ0 ) ∩ T ∗ Ωj ⊂ exp tHa0 (K(f )) ∩ T ∗ Ωj , for all t ≥ 0. Then, for any j ∈ {0, 1, . . , r}, ψj , ψ˜j ∈ C0∞ (Ωj ) with ψ˜j ψj = ψj , and any α ∈ C0∞ (IRn ), we write, Uj ψj e−itP/h ϕ0 = α(hDx )ψ˜j (x)Uj ψj e−itP/h ϕ0 + (1 − α(hDx ))Uj ψj e−itP/h ϕ0 , and therefore, if α(ξ) = 1 in a sufficiently large compact set, Uj ψj e−itP/h ϕ0 = α(hDx )ψ˜j (x)Uj ψj e−itP/h ϕ0 + O(h∞ ).
Let A be a U-twisted PDO on L2 (IRn ; H) of degree µ, where U is the regular covering defined in Chapter 2. Then, for any integers k, such that k + ≥ µ/m, the operator (P˜ +i)−k A(P˜ +i)− is a U-twisted h-admissible operator. Proof – We first consider the case k = 0. ,r is an arbitrary partition of unity with χν ∈ Cd∞ (Ων ), χ ˜ ν (x) + ζ(x)W (x). 3. Let j ∈ {0, . . , r} and ψj ∈ Cd∞ (Ωj ) be fixed. ,r of IRn with χν ∈ Cd∞ (Ων ), and there exists χ ∞ χ ˜ χ χ χ ˜ χ Cd (Ων ) with ν ν = ν (ν = 0, . .