The asymptotics of inhomogeneous factored Cauchy problems


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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