Abstract The rectilinear or rod model of neutron transport consists of a segment which is assumed to be capable of transporting particles. Such particles (neutrons) can move only to the right or to the left on the segment at a constant speed, and suffer scattering, capture and fission. We have shown that the time-eigenvalue spectrum of this model is made of a finite, non-void set of real eigenvalues, plus infinitely many complex eigenvalues, distributed in pairs of complex conjugates. A simple asymptotic formula for the high-order complex eigenvalues has been obtained. As an application, we have considered the following initial value problem: find the outgoing density of neutrons, at one end of the rod, at any time after the injection of a neutron pulse at the same end. A complete series expansion of the outgoing density has been given. The successive time-derivatives of this expansion show certain characteristic discontinuities which are due to the motion, from one end of the rod to the other, of discont...