Linear adiabatic dynamics generated by operators with continuous spectrum. I

We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε∂tψ = A(t)ψ, A(t) = A0 + V (t), ψ(0) = ψ0, in the limit ε → 0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ0 is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t), we derive a formal asymptotic solution of the problem whose leading order has an explicit representation. A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical pseudo-differential operator M = M(t, iε∂t) with compact operator-valued symbol M(t, E) = V1(t)(A0 − EI)−1V2(t) , V (t) = V2(t)V1(t), and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schrödinger equation on the axis with the δ-function potential: A(t) = −∂xx + α(t)δ(x).