Scattering by infinitely rising one-dimensional potentials

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.