## Linear adiabatic dynamics generated by operators with continuous spectrum. I.(English)Zbl 1152.35336

Summary: We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation $i\varepsilon\partial_t\psi= A(t)\psi,\quad A(t)= A_0+ V(t),\quad \psi(0)= \psi_0,$ in the limit $$\varepsilon\to 0$$. A case of special interest is when the operator $$A(t)$$ has continuous spectrum and the initial data $$\psi_0$$ 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 $${\mathcal M}= M(t,i\varepsilon\partial_t)$$ with compact operator-valued symbol $$M(t,E)= V_1(t)(A_0- EI)^{-1}V_2(t)$$, $$V(t)= V_2(t)V_1(t)$$, 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 $$\delta$$-function potential: $$A(t)=-\partial_{xx}+ \alpha(t)\delta(x)$$.

### MSC:

 35G10 Initial value problems for linear higher-order PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B25 Singular perturbations in context of PDEs