Invariant subspaces of analytic multiparticle Hamiltonians

The quantum mechanics of n particles interacting through analytic two-body interactions can be formulated as a problem of functional analysis on a Hilbert space $\text{G}$ consisting of analytic functions. On $\text{G}$, there is an Hamiltonian H with resolvent R(λ). These quantities are associated with families of operators H(?) and R(λ, ?) on $\text{L}$, the case ? = 0 corresponding to standard quantum mechanics. The spectrum of H(?) consists of possible isolated points, plus a number of half-lines starting at the thresholds of scattering channels and making an angle 2? with the real axis.Assuming that the two-body interactions are in the Schmidt class on the two-particle space $\text{G}$, this paper studies the resolvent R(λ, ?) in the case ? ≠ 0. It is shown that a well known Fredholm equation for R(λ, ?) can be solved by the Neumann series whenever ¦λ¦ is sufficiently large and λ is not on a singular half-line. Owing to this, R(λ, ?) can be integrated around the various half-lines to yield bounded idempotent operators P_p(?) (p = 1, 2,…) on $\text{L}$. The range of P_p(?) is an invariant subspace of H(?). As ? varies, the family of operators P_p(?) generates a bounded idempotent operator P_p on a space $\text{G}$. The range of this is an invariant subspace of H. The relevance of this result to the problem of asymptotic completeness is indicated.