Boundary value problems in time for wave equations on R N

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