Cauchy problem for degenerating linear differential equations and averaging of approximating regularizations

V. Zh. Sakbaev

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 43 (2012), pp. 3-172

Résumé

In this work, we consider the Cauchy problem for the Schrödinger equation. The generating operator $\mathbf L$ for this equation is a symmetric linear differential operator in the Hilbert space $H=L_2(\mathbb R^d)$, $d\in\mathbb N$, 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 $\{\mathbf L_n\}$ of the generating operator $\mathbf L$. When there are no convergent sequences of regularized solutions, we study the convergence of the corresponding sequence of the regularized density operators.