Perturbation theory for eigenvalues and resonances of Schrödinger hamiltonians

AbstractSuppose that e2ϵ|x|V ∈ ReLP(R3) for some p > 2 and for g ∈ R, H(g) = − Δ + g V, H(g) = −Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is λ(g) ↗6 0 as g ↘5 g0 > 0. We find a series expansion in powers of (g − g0)12, λ(g) = ∑n = 2∞ an(g − g0)n2 whose values for g < g0 correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues