M. W. Smiley and A. M. Fink

Copyright © 1990 M. W. Smiley and A. M. Fink. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let Lλ denote the linear operator associated with the radially symmetric form of the wave operator ∂t2−Δ+λ together with the side conditions of decay to zero as r=‖x‖→+∞ and T-periodicity in time. Thus Lλω=ωtt−(ωrr+N−1rωr)+λω, when there are N space variables. For δ,R,T>0 let DT,R=(0,T)×(R,+∞) and Lδ2(D) denote the weighted L2 space with weight function exp(δr). It is shown that Lλ is a Fredholm operator from dom(Lλ)⊂L2(D) onto Lδ2(D) with non-negative index depending on λ. If [2πj/T]2<λ≤[2π(j+1)/T]2 then the index is 2j+1. In addition it is shown that Lλ has a bounded partial inverse Kλ:Lδ2(D)→Hδ1(D)⋂Lδ∞(D), with all spaces weighted by the function exp(δr). This provides a key ingredient for the analysis of nonlinear problems via the method of alternative problems.