The scope of the function theoretic approach for equations permitting a separation of variables

Abstract

The motivation to continue analytically solutions of partial differential equations, and determine the location and type of singularities is generated by both modern and classical physical theories. In the area of modern physics, problems which occur in the scattering of particles, using either the potential scattering model [16, 28, 291, or the field theoretic model [12], require the analytic continuation of solutions of partial differential equations, and a large literature has been established concerning these problems. In classical physics, a revived interest has been shown recently concerning analytical methods. Indeed, as has been pointed out by Millar [27], the study of magnetic and gravitational anomalies has been enhanced by the use of these techniques; see in this regard the works of the Soviet researchers, Strakhov [30], and Voskoboinikov [34]. With this in mind, it has been our intent to organize a detailed research program concerning the analytic structure of solutions of partial differential equations. We have concentrated on elliptic partial differential equations of second order, since these are the types which have to date occurred in the applications. However, similar problems concerning the solutions of para- bolic and hyperbolic equations arise. The parabolic and hyperbolic equations have been investigated in detail by Widder [35] and by Leray [25, 261 respectively. The elliptic equations have been investigated by Bergman [l-3],