Integral representations for products of Airy functions and their application for analysis of the Green’s function for a particle in a uniform static field

Abstract Representations for products of two Airy functions with different complex arguments in the form of one-dimensional contour integrals are obtained. These representations are used for analysis of the Green’s function for a charged particle in a uniform static electric field. The integral relation between the stationary and time-dependent Green’s functions is discussed in the sense of its analytical properties for complex energy and field strength. It is shown that the Green’s function can be divided into analytic and non-analytic parts with respect to the field strength near its zero.