ИСТИНА

We consider the spectral problem for the one-dimensional Shroedinger operator with a potential having a localized perturbation near the fixed point. The width of the perturbation tends to zero more slowly than the semiclassical parameter h. We consider the width to be order of the root h. After defining a fast variable as (x-x0)/sqrt(h) we have a case with different small parameters before second derivatives by the fast and slow variables. So the geometric object is a one-dimensional bundle over a circle. Layers are due to the presence of a fast variable. We have to modify the Maslov’s canonical operator to construct an asymptotics of eigenfunctions. We considered two cases of the position of the total energy: larger than the maximum of localized perturbation and less than the maximum of localized perturbation but larger than the minimum of the potential.