Resolvent estimates for a certain class of Schrödinger operators with exploding potentials

Let H = −Δ + VE(¦x¦)+ V(x) be a Schrödinger operator in Rn. Here VE(¦x¦) is an “exploding” radially symmetric potential which is at least C2 monotone nonincreasing and O(r2) as r → ∞. V is a general potential which is short range with respect to VE. In particular, VE  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = limr → ∞ VE(r) and R(z) = (H − z)−1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset N⊆(Λ, ∞), in a suitable operator topology B(L, L∗). And L ⊆ L2(Rn) is a weighted L2-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing.