Asymptotic behavior at infinity of the Green function of a class of systems including wave propagation in crystals
The first order symmetric hyperbolic system can be written in the form \[ (1)\quad E(x)D_ tu-\sum A_ jD_ ju=\tilde f(t,x), \] where \(iD_ t=\partial /\partial t\), \(iD_ j=\partial /\partial x_ j\), E(x) is a positive definite Hermitian matrix, \(A_ j\) are \(m\times m\) Hermitian matrices. Multiplying (1) by \(E^{-1}\) reduces it to the form \(D_ tu- \Lambda^ 0u=f(t,x).\) If f(x,t) is given by \[ -f(t,x)=e^{i\lambda t}f(x),\quad \lambda \in {\mathbb{R}}\setminus \{0\}, \] then the solution u(t,x) also must be of the form \(u(x,t)=e^{i\lambda t}v(x,\lambda),\) and v(x,\(\lambda)\) solves the eigenvalue problem \((2)\quad \Lambda^ 0v- \lambda v=f(x).\) The Green function G(x,\(\xi)\) is defined by the inverse Fourier transform \(G(x,\xi)={\mathcal F}^{-1}[(\Lambda^ 0-\xi I)^{- 1}],\) but it is useful to think of it as the elementary solution of (2), i.e. \((\Lambda^ 0-\lambda I)G(x,\lambda)=\delta (x)\cdot I.\) The author develops representation of Green's function in terms of the slowness integral in the style of C. Wilcox and J. R. Schulenberger. (The slowness surface is the polar inverse with respect to the unit sphere of the wave surface.) The author proceeds to develop a quite impressive list of results concerning estimates for Green's function, properties of the slowness surface, which for electromagnetic waves in crystals is called the Fresnel surface, existence of stationary points, and asymptotic properties of the Green function. Some of the results are quite lengthy and technically difficult because of consideration of special cases, odd or even spatial dimensions and a selection of different assumptions concerning the conditions on the slowness surface S, such as: \((S_ i)\) stating that the set of all algebraic singularities forms an (n-d) smooth manifold, with \(d\geq (n+1)/2\). Small misprints mar the presentation. On page 1 the Hermitian matrices are not constant, \(\Delta\) should be \(\Lambda\), etc., but these are minor points hardly detracting from a technically difficult article.
scientific article; zbMATH DE number 3942068