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

Research paper by V. Zh. Sakbaev

Indexed on: 25 Jan '16Published on: 25 Jan '16Published in: Journal of Mathematical Sciences



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(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 \( \left\{{e}^{-i{\mathbf{L}}_nt},\ t>0\right\} \) 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.