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

Matania Ben-Artzi*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Scopus citations

Abstract

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) 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.

Original languageEnglish
Pages (from-to)259-274
Number of pages16
JournalIsrael Journal of Mathematics
Volume40
Issue number3-4
DOIs
StatePublished - Sep 1981
Externally publishedYes

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