A limiting absorption principle for Schrödinger operators with spherically symmetric exploding potentials

Let H=-Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely, V(r)=V _S(r)+V _L(r), where V _S(r) is short-range and the exploding part V _L(r) satisfies the following assumptions: (a) Λ=lim sup _r→∞ V _L(r)<∞ (but Λ=-∞ is possible). Denote Λ⁺= max(Λ,0). (b)V _L(r)∈C ^2k (r ₀, ∞) and, with some δ>0 such that 2 kδ>1: (d/dr) ^j V _L(r) · (Λ⁺-V _L(r))^-1=O(r ^jδ) as r → ∞, j=1, ..., 2 k. (c) ∫ _r0 ^∞ dr|V _L(r|^1/2 dr|V _L(r)|^1/2=∞. (d) (d/dr)V _L(r)≦0. Under these assumptions a limiting absorption principle for R(z)=(H-z)^-1 is established. More specifically, if K ⊆C ⁺={zIm z≧0} is compact and K ∩ (-∞, Λ]=Ø then R (z) can be extended as a continuous map of K into B (Y, Y*) (with the uniform operator topology), where Y ⊆L ²(R ⁿ) is a weighted-L ² space. To ensure uniqueness of solutions of (H-z)u=f, z ∈K, a suitable radiation condition is introduced.