Positive resonances and the limiting amplitude principle (Q1085317)

Suppose that the Cauchy problem \[ d^ 2w/dt^ 2+Aw=e^{-iwt}f,\quad t>0,\quad w|_{t=0}=dw/dt|_{t=0}=0 \] (A is a positive selfadjoint operator in the separable Hilbert space H, \(w>0\), \(f\in H\), \(w: (0,t)\to H)\) has a unique solution and that conditions are satisfied in order that \(w=e^{-iwt}u+o(1)\), where u satisfies (3) \(\bar Au=e^{- iwt}u+f\). In the presence of resonance for \(\lambda\geq 0\), (3) may not be satisfied. The author gives a suitable definition for the positive resonance so that relations analogous to (3) remain valid.