Maslov complex germ and semiclassical contracted states in the Cauchy problem for the Schrödinger equation with delta potential

We describe the semiclassical asymptotic behavior of the solution of the Cauchy problem for the Schrödinger equation with a delta potential localized on a surface of codimension 1. The Schrödinger operator with a delta potential is defined using the theory of extensions and is given by the boundary conditions on this surface. The initial data are selected as a narrow peak, which is a Gaussian packet localized in a small neighborhood of the point. To construct the asymptotics, we use the Maslov complex germ method. We describe the reflection of the complex germ from the carrier of the delta potential.