Spectral representations and asymptotic wave functions for long-range perturbations of the d'Alembert equation (Q791766)

This paper is devoted to the study of the asymptotic behaviour \(t\to \infty\) of the acoustic wave w(t,x) in inhomogeneous media: \[ \frac{\partial^ 2}{\partial t^ 2}w(t,\cdot)+Lw(t,\cdot)=0, \] where the selfadjoint operator L in \(L^ 2({\mathbb{R}}^ n)\) (\(n\geq 2)\) is a long-range perturbation of -\(\Delta\), defined by \[ Lu=- \sum^{n}_{j,k=1}\frac{\partial}{\partial x_ j}(a_{jk}(x)\frac{\partial}{\partial x_ k}u). \] On the basis of the spectral representation theory, the asymptotic wave function corresponding to w(t,x) is constructed from initial data and determined as a modified diverging spherical wave. This is an extension of the result of \textit{K. Mochizuki} [J. Math. Soc. Japan 34, 143-171 (1982; Zbl 0465.76073)]. Most of the contents are concerned with the establishment of spectral representations for L, which is carried out by use of rigorous approximate phase function.