ИСТИНА

Given a one-dimensional Schr\”{o}dinger operator with the semi-classical parameter $h$, whose potential has a single well. It is well known that its spectrum modulo small terms is given by the values of the Hamiltonian function on closed invariant curves $\Lambda_n$ satisfying the Bohr-Sommerfeld quantization rule. Given some other closed curve $\Gamma$ (also satisfying the Bohr-Sommerfeld rule) endowed with the measure, let K_{\Gamma$ be the associated Maslov’s canonical operator. We prove that the function $K_{\Gamma}A$ (for any smooth $A$) modulo small terms in $h$ belongs to some subspace of eigenfunctions of the above Schr\”{o}dinger operator. Namely, this subspace consists of the eigenfunctions corresponding to the curves $\Lambda_n$ having the non-empty intersection with the neighborhood of $\Gamma$. We discuss the application of the mentioned result to the so-called quantum adiabatic theorem in a semi-classical setting. This theorem was stated as a conjecture by Neishtadt and Kuksin. In this theorem, one considers he non-stationary Schr\”{o}dinger equation with the single-well one-dimensional potential slowly varying with time. In this problem, one has two small parameters, semi-classical h, and adiabatic $\varepsilon$. As an initial condition, we take a Schr\”{o}dinger operator eigenfunction corresponding to the time $t=0$. Then the theorem describes the proximity of the solution of such Cauchy problem to some subspace of Schr\{o}dinger equation eigenfunctions corresponding to the time $t=1/\varepsilon$. This theorem is, actually, the semi-classical analog of the fact from classical perturbation theory about the action variables adiabatic invariance.