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Semiclassical resolvent estimates for two and three-body Schrödinger operators. (English) Zbl 0711.35096
Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1989, Exp. No. 18, 10 p. (1989).
Resolvent estimates in the semiclassical limit for generalized three-body Schrödinger operators $$H=-h^ 2\Delta +V(x)$$ on $${\mathbb{R}}^ n$$ are established. Here V(x)$$\equiv \sum_{a\in A}V_ a(\pi^ ax)$$ with $$\{\pi^ a\}_{a\in A}$$ a family of orthogonal projections on vector subspaces $$X^ a$$ of $${\mathbb{R}}^ n.$$
Two-body Schrödinger operators are considered when the energy level tends to zero. A method of C. Gérard and A. Martinez [C. R. Acad. Sci., Paris, Sér. I 306, No.3, 121-123 (1988; Zbl 0672.35013)] is used which is based on Mourre’s commutator technique and on the construction of a conjugate operator for H by quantizing a classical escape function which increases along the classical flow.

##### MSC:
 35P25 Scattering theory for PDEs 35J10 Schrödinger operator, Schrödinger equation 81U10 $$n$$-body potential quantum scattering theory 35S05 Pseudodifferential operators as generalizations of partial differential operators
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