On a variant of commutator estimates in spectral theory

Completeness is proved for systems of two or three quantum particles. The proof is based on the following statement on operators in a Hilbert space. If an operator A is bounded with respect to a self-adjoint operator H and Re((H – )f, Af) Bf², then B is smooth with respect to H.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 29–36, 1987.