Perturbation of embedded eigenvalues in the generalizedN-body problem

We discuss the perturbation of continuum eigenvalues without analyticity assumptions. Among our results, we show that generally a small perturbation removes these eigenvalues in accordance with Fermi's Golden Rule. Thus, generically (in a Baire category sense), the Schrodinger operator has no embedded non-threshold eigenvalues. I. Introduction It is well known (R-Sl) that a one-body Schrodinger operator — A + V(x), where V is sufficiently well behaved at infinity, cannot have eigenvalues λ embedded in the continuous spectrum (except possibly at threshold, λ = 0). The situation is quite different in the JV-body problem where continuum eigenvalues not only can exist, but do indeed exist in important physical situations: The operator Ho = -Δ1-Δ2-2/\x1\-2/\x2\ in L2{U6) (describing the Helium atom without electronic repulsion) has eigenvalues embedded in the continuous spectrum. While this example has an obvious symmetry, such symmetry is not necessary for the existence of embedded eigenvalues. An example in (F-H-HO-HO) can be modified to produce an embedded eigenvalue where no symmetry is apparent. In (Howl, 2) and (SI), analyticity assumptions are made which allow the treatment of embedded eigenvalues using the perturbation theory developed for use with isolated eigenvalues. The major idea in this theory is that when a small perturbation βW is added to the Schrodinger operator H, the continuum eigenvalue Eo turns into a "resonance," E0(β), which, while not necessarily an eigenvalue of H + βW, is a pole in the analytic continuation of certain matrix elements (φ,(H + βW — z)~ιφ) of the resolvent. The function E0(β) is analytic in β for \β\ small. E0(β) has an imaginary part which appears first to second order in β: τ d2E0(β)