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

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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 = L 2(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 {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.

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  1. Moscow Institute of Physics and Engineering, Moscow, Russia

    V. Zh. Sakbaev

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 43, Partial Differential Equations, 2012.

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Sakbaev, V.Z. Cauchy Problem for Degenerating Linear Differential Equations and Averaging of Approximating Regularizations. J Math Sci 213, 287–459 (2016). https://doi.org/10.1007/s10958-016-2719-z

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