Cauchy Problem for Degenerating Linear Differential Equations and Averaging of Approximating Regularizations

In this work, we consider the Cauchy problem for the Schrödinger equation. The generating operator L for this equation is a symmetric linear differential operator in the Hilbert space H = L2(ℝd), d ∈ ℕ, degenerated on some subset of the coordinate space. To study the Cauchy problem when conditions of existence of the solution are violated, we extend the notion of a solution and change the statement of the problem by means of such methods of analysis of ill-posed problems as the method of elliptic regularization (vanishing viscosity method) and the quasisolutions method. We investigate the behavior of the sequence of regularized semigroups (Formula presented.) depending on the choice of regularization {Ln} of the generating operator L. When there are no convergent sequences of regularized solutions, we study the convergence of the corresponding sequence of the regularized density operators. © 2016, Springer Science+Business Media New York.