Asymptotics of solutions of the Schrödinger equation with a two-dimensional quadratic inverted potential and localized right-hand side

We consider the Schrödinger equation with potential $$-x_1^2-bx_2^2$$ and with a localized right-hand side. In this problem, the condition that the rays of the Hamiltonian system leave any finite region of the configuration space in finite time is not satisfied. Because of this, it is not possible to use the well-known algorithm for constructing an asymptotic solution in the form of a canonical operator on a certain Lagrangian manifold. We construct a solution to the equation in the form of an integral, express its asymptotics via known functions, and study the dependence of this solution on the parameter $$b$$ .