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).