**Abstract: ** I will consider the Dirichlet Laplacian $H_\theta$ in a twisted
waveguide with rotational angle $\theta$ depending on the
longitudinal variable.
First, I will discuss the case where the derivative $\theta'$
can be written as $\beta - \epsilon$ with a constant $\beta >
0$, and a decaying function $\epsilon \geq 0$, and will discuss the
asymptotic distribution of the discrete spectrum of $H_\theta$
near the bottom of its essential spectrum.
Further, I will show that the wave operators for the operator pair
$(H_\theta_1, H_\theta_2)$ exist and are complete, provided that
the derivative $\theta'_1 - \theta'_2$ decays fast enough at infinity.
Finally, I will assume that $\theta' = \beta - \epsilon$ with a
real constant $\beta$, and a real function
$\epsilon$ which decays fast enough at infinity. Using appropriate
Mourre estimates, I will show
that the singular continuous spectrum of $H_\theta$ $ is empty.
The talk will be based on joint works with Ph. Briet, H. Kovarik,
and E. Soccorsi.