Generic simplicity of resonances in obstacle scattering

We show that all resonances in Dirichlet obstacle scattering (in $\mathbb{C}$ in odd dimensions and in the logarithmic cover of $\mathbb{C}\setminus \{0\}$ in even dimensions) are generically simple in the class of obstacles with $C^k$ (and $C^{\mathrm{\infty }}$) boundaries, $k\ge 2$.