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

Abstract

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(ℝ^d), 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 \( \left\{{e}^{-i{\mathbf{L}}_nt},\ t>0\right\} \) depending on the choice of regularization {L_n} 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.