  We consider the reflection-transmission problem in a waveguide with obstacle.
At certain frequencies, for some incident waves, intensity is perfectly
transmitted and the reflected field decays exponentially at infinity. In this
work, we show that such reflectionless modes can be characterized as
eigenfunctions of an original non-selfadjoint spectral problem. In order to
select ingoing waves on one side of the obstacle and outgoing waves on the
other side, we use complex scalings (or Perfectly Matched Layers) with
imaginary parts of different signs. We prove that the real eigenvalues of the
obtained spectrum correspond either to trapped modes (or bound states in the
continuum) or to reflectionless modes. Interestingly, complex eigenvalues also
contain useful information on weak reflection cases. When the geometry has
certain symmetries, the new spectral problem enters the class of
$\mathcal{PT}$-symmetric problems.
