"The asymptotics of inhomogeneous factored Cauchy problems" by Gisèle Ruiz Goldstein, Jerome A. Goldstein et al.
 

The asymptotics of inhomogeneous factored Cauchy problems

Abstract

Scattering theory tells how solutions of one abstract Schrödinger equation (of the form i(du/dt) = Hu with H = H*) are asymptotic to solutions of another (in principle simpler) abstract Schrödinger equation. We extend this theory to inhomogeneous problems of the form i(du)/(dt) = Hu + h(t), with special emphasis on factored equations of the form ΠNj=I((d/dt) - iAj)u(t) = h(t), where AI, . . . , AN are commuting selfadjoint operators. As a special case, corresponding to N = 4 and two-space scattering, we conclude that every solution u(., t) of the inhomogeneous elastic wave equation in the exterior of a bounded star shaped obstacle is of the form u = v + w + z, where v(., t) solves the free (homogeneous) elastic wave equation with no obstacle, w(., t) is determined by the (rather general) inhomogeneity, and z(., t) = o(1) as t → ±∞. Some of the results are presented in a more general Banach space context.

Publication Title

Asymptotic Analysis

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