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

Abstract

LetH=−Δ+V(r) be a Schrödinger operator with a spherically symmetric exploding potential, namely,V(r)=V _S(r)+V _L(r), whereV _S(r) is short-range and the exploding partV _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 someδ>0 such that 2kδ>1: (d/dr)^j V _L(r) · (Λ⁺−V _L(r))⁻¹=O(r ^jδ) asr → ∞,j=1, ..., 2k. (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 forR(z)=(H−z)⁻¹ is established. More specifically, ifK ⊆C ⁺={zImz≧0} is compact andK ∩ (−∞, Λ]=Ø thenR (z) can be extended as a continuous map ofK intoB (Y, Y*) (with the uniform operator topology), whereY ⊆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.