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We consider the solution $u$ to the one-dimensional parabolic Anderson modelwith homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift anda time-independent potential bounded from above. Under ergodicity andindependence conditions we derive representations for both the quenchedLyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponentsfor {\it all} $p \in (0, \infty).$ These results enable us to prove theheuristically plausible fact that the $p$-th annealed Lyapunov exponentconverges to the quenched Lyapunov exponent as $p \downarrow 0.$ Furthermore,we show that $u$ is $p$-intermittent for $p$ large enough. As a byproduct, wecompute the optimal quenched speed of the random walk appearing in theFeynman-Kac representation of $u$ under the corresponding Gibbs measure. Inthis context, depending on the negativity of the potential, a phase transitionfrom zero speed to positive speed appears.

Lyapunov exponents for the one-dimensional parabolic Anderson model with drift