Ulrich bundles on smooth toric threefolds with Picard number $2$

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number$2$, namely $\mathbb{P}({\mathcal{O}}_{{\mathbb{P}}^2}(a_0)\oplus {\mathcal{O}}_{{\mathbb{P}}^2}(a_1))$. We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from ${\mathbb{P}}^2$. As a consequence, we also show that these varieties are Ulrich wild.