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

Let $\text{H = ?Δ + V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)+ V(x)}$ be a Schrödinger operator in Rⁿ. Here $\text{V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)}$ 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 V_E. In particular, V_E  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = lim_{r → ∞}V_E(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 $\text{N}$?(Λ, ∞), in a suitable operator topology $\text{B(L, L}{}^1\text{)}$. 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.