On the spectral theory of Rayleigh's piston. I. The discrete spectrum

The classic problem of Rayleigh's piston is reconsidered in the light of modern transport theory. The velocity relaxation of a one-dimensional ensemble of test particles immersed in a similar heat bath at arbitrary mass ratio gamma is investigated, and a new reduction of the integral collision operator to an infinite-order differential expansion is obtained, which is more tractable than the conventional expression in powers of the mass ratio. In this way it is proved that the number of nonzero discrete eigenvalues of the Rayleigh collision operator is always bounded for finite mass ratio, being in fact zero for the special case gamma =1. By truncation of the collision operator expansion, a tentative bound is then obtained which suggests (subject to an unproved positive-definiteness condition) that the emptiness of the discretum actually extends over at least the mass ratio region (3( nth root 2-1))-1< gamma <3( nth root 2-1). The limit gamma to 0 may be studied and gives a novel approach to Rayleigh's original solution under conditions of brownian motion.