  Infinitely rising one-dimensional potentials constitute impenetrable barriers
which reflect totally any incident wave. However, the scattering by such kind
of potentials is not structureless: resonances may occur for certain values of
the energy. Here we consider the problem of scattering by the members of a
family of potentials $V_a(x)=-{\rm sgn}(x)\,|x|^a$, where sgn represents the
sign function and $a$ is a positive rational number. The scattering function
and the phase shifts are obtained from global solutions of the Schr\"odinger
equation. For the determination of the Gamow states, associated to resonances,
we exploit their close relation with the eigenvalues of the
$\mathcal{PT}$-symmetric Hamiltonians with potentials
$V_a^{\mathcal{PT}}(x)=-{\rm i}\,{\rm sgn}(x)\,|x|^a$. Calculation of the time
delay in the scattering at real energies is used to characterize the
resonances. As an additional result, the breakdown of the
$\mathcal{PT}$-symmetry of the family of potentials $V_a^{\mathcal{PT}}$ for
$a<3$ may be conjectured.
