The primitive ideal space of the partial-isometric crossed product of a system by a single automorphism

Let $(A,\alpha)$ be a system consisting of a $C^*$-algebra $A$ and an automorphism $\alpha$ of $A$. We describe the primitive ideal space of the partial-isometric crossed product $A\times_{\alpha}^{\textrm{piso}}\mathbb{N}$ of the system by using its realization as a full corner of a classical crossed product and applying some results of Williams and Echterhoff.