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.