The singular terms in the fundamental matrix of crystal optics †

The theory of partial differential equations has evolved in close concurrence with specific examples coming from physics. A prominent case in the history of science was the propagation of light in crystals investigated by such great physicists as Huygens, Fresnel, Biot, Brewster and Laplace. In particular, the prediction of conical refraction in biaxial crystals by Hamilton in 1832 and its experimental verification shortly thereafter by Lloyd fascinated physicists and mathematicians alike, triggering a long-lasting search for a mathematical framework encompassing and explaining such phenomena. This search was advanced by scientists like Navier, Cauchy, Green, Stokes, Lamé, Kovalevskaya, Volterra, Grünwald, Zeilon, Herglotz, etc., cf. the historical account in Gårding (1989), culminating in the modern theory of hyperbolic differential operators, initiated by Hadamard, and elaborated by, among others, Petrovsky, Riesz, Gårding and Hörmander. This theory allows to analyse the singularities and lacunas of solutions of hyperbolic operators by means of localizations and microlocal analysis, i.e. wavefront sets, see Petrovsky (1945), Hörmander (1969, 1983a,b), Atiyah et al. (1970, 1973) and Gårding (1984). Parallel to this qualitative study, several methods were devised for the calculation, or representation by integrals, of fundamental solutions of differential operators by Laplace, Poisson, Fourier, Stokes, Zeilon, Fredholm, Herglotz and others. Placed on a sound mathematical basis by Schwartz’s invention of distribution theory around 1950, such integral representations of fundamental solutions were further generalized by Petrovsky, Leray, Gel’fand, Shilov, etc., and