Propagation estimates for \(N\)-body Schrödinger operators. (English) Zbl 0760.35035

This paper proves time decay estimates of the form \([B(t) \exp(-itH) f(H) <x>^{-s'} =\text{O}(t^{-s})]\), where \(H\) is a suitable long-range \(N\)-body Schrödinger operator, \(B(t)\) is a pseudo-differential operator and \(f\) is smooth with compact support. These results imply versions of minimal and large velocity estimates of Sigal and Soffer [I. M. Sigal and A. Soffer, e.g.: Ann. Math., II. Ser. 126, 35-108 (1987; Zbl 0646.47009)], as well as estimates for the “free channel region” which were known in very special cases. The author notes that the results can be extended to time-dependent Hamiltonians. The verification of the author’s hypothesis in concrete examples relies on a partition of unity adapted to the cluster decomposition due to G. M. Graf [Commun. Math. Phys. 132, No. 1, 73-101 (1990; Zbl 0726.35096)].


35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81U05 \(2\)-body potential quantum scattering theory
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