Generalized Coherent States of a Particle in a Time-Dependent Linear Potential

We derive, with an invariant operator method and unitary transformation approach, that the Schrödinger equation with a time-dependent linear potential possesses an infinite string of shape-preseving wave-packet states |φ_α,λ>(t)> having classical motion. The qualitative properties of the invariant eigenvalue spectrum (discrete or continuous) are described separately for the different values of the frequency ω of a harmonic oscillator. It is also shown that, for a discrete eigenvalue spectrum, the states |φ_α,n>(t)> could be obtained from the coherent state |φ_α,0>(t).