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Modified wave operators and Stark Hamiltonians. (English) Zbl 0766.35033

On prouve l’existence et la complétude des opérateurs d’onde modifiés en mécanique quantique, en présence d’un champ électrique constant. Plus exactement, soient \(H_ 0=-\Delta/2-x_ 1\), \(H=H_ 0+V\) des opérateurs auto-adjoints sur \(L^ 2(\mathbb{R}^ n)\). On suppose \(V=V_ L+V_ S\), où \(V_ L\) est une perturbation du type “longue portée”, \(V_ L\in{\mathcal C}^ \infty(\mathbb{R}^ n;\mathbb{R})\), telle qu’il existe \(\varepsilon>0\) et les constantes \(C_ \alpha>0\), \(\alpha\in\mathbb{R}^ n\), telles que \[ | D^ \alpha V_ L(x)|\leq C_ \alpha\langle x_ 1\rangle^{-|\alpha|/2-\varepsilon}, \qquad D^ \alpha V_ L(n)=o(1) \quad\text{ si } | n|\to \infty,\quad \alpha\in\mathbb{N}^ n, \] tandis que \(V_ S\) est une perturbation du type “courte portée”, opérateur symétrique, \(H_ 0\)-compact et tel que \[ \int_ 1^ \infty \| F(x_ 1>r^ 2)V_ S(H_ 0+i)^{-1}\| dr<\infty, \] où \(F(\cdot)\) est la fonction caractéristique de l’ensemble indiqué. Sous ces hypothèses on construit une fonction \(X=X(\xi_ 1,\dots,\xi_ n,t)\), telle que si \(X_ D(t)=X(D_ 2,\dots,D_ n,t)\), les opérateurs d’onde modifiés \(W_ D^ \pm =\text{s-lim}_{t\to\mp\infty} \exp(it H)\exp(-it H_ 0)\exp(-i X_ D(t))\) existent et sont complets. On donne aussi une condition nécessaire et suffisante afin qu’il existe les opérateurs d’onde de Møller.

MSC:

35P25 Scattering theory for PDEs
47A40 Scattering theory of linear operators
81U10 \(n\)-body potential quantum scattering theory
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