ИСТИНА

We consider a Cauchy problem for a one-dimensional Schrodinger equation in semi-classical approximation (with respect to the small parameter $h$). The potential is assumed to vary slowly with respect to another small parameter $\epsilon$. Let us take an eigenfunction of the Schrodinger operator for time $t=0$ generated by a closed trajectory of the classical Hamiltonian. We take it as an initial condition for the Cauchy problem and study the evolution of the solution for times $t\in[0,\varepsilon]$. Using the method of Maslov's canonical operator [1] we estimate its closeness to the eigenfunctions of the Schrodinger operator for time $t$. This result is a semi-classical analog of the fact that the classical action is an adiabatic invariant [2]. This work was done together with S.Dobrokhotov, S.Kuksin and A.Neishtadt, and was supported by RFBR-CRNS Grant No. 17-51-150006.