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

Abstract

LetH=−Δ+V(r) be a Schrodinger 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) 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]=O 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.