  Many active matter systems are known to perform L\'{e}vy walks during
migration or foraging. Such superdiffusive transport indicates long-range
correlated dynamics. These behavior patterns have been observed for
microswimmers such as bacteria in microfluidic experiments, where Gaussian
noise assumptions are insufficient to explain the data. We introduce
\textit{active L\'evy swimmers} to model such behavior. The focus is on ideal
swimmers that only interact with the walls but not with each other, which
reduces to the classical L\'evy walk model but now under confinement. We study
the density distribution in the channel and force exerted on the walls by the
L\'evy swimmers, where the boundaries require proper explicit treatment. We
analyze stronger confinement via a set of coupled kinetics equations and the
swimmers' stochastic trajectories. Previous literature demonstrated that
power-law scaling in a multiscale analysis in free space results in a
fractional diffusion equation. We show that in a channel, in the weak
confinement limit active L\'evy swimmers are governed by a modified Riesz
fractional derivative. Leveraging recent results on fractional fluxes, we
derive steady state solutions for the bulk density distribution of active
L\'evy swimmers in a channel, and demonstrate that these solutions agree well
with particle simulations. The profiles are non-uniform over the entire domain,
in contrast to constant-in-the-bulk profiles of active Brownian and
run-and-tumble particles. Our theory provides a mathematical framework for
L\'evy walks under confinement with sliding no-flux boundary conditions and
provides a foundation for studies of interacting active L\'evy swimmers.
