Principle of limiting absorption for N-body Schrödinger operators — A remark on the commutator method

Abstract

We consider a N-body Schrödinger operator H=H ₀+V. The interaction V is given by a sum of pair potentials V _jk(y)(=V ^s_jk +V ^l_jk ), y ∈ R³. We assume that: V ^s_jk =O(|y|^-(1+p)), p>0, as |y| → ∞ for the short-range part V ^s_jk ; $\partial _y^\alpha V_{jk}^l = 0(|y|^{ - (|\alpha | + p)} ),{\text{ }}0 \leqslant {\text{ }}|a| \leqslant 1,{\text{ }}as |y| \to \infty$ for the long-range part V ^l_jk . Under this assumption, we prove the principle of limiting absorption for H. The obtained result is essentially as good as those obtained in the two-body case. The proof is done by a slight modification of the remarkable commutator method due to Mourre.