Spectral singularities of odd-$PT$-symmetric potential

We describe one-dimensional stationary scattering of a two-component wave field by a non-Hermitian matrix potential which features odd-$PT$ symmetry, i.e., symmetry with $(PT)^2=-1$. The scattering is characterized by a $4\times 4$ transfer matrix. The main attention is focused on spectral singularities which are classified into two types. Weak spectral singularities are characterized by zero determinant of a diagonal $2\times 2$ block of the transfer matrix. This situation corresponds to the lasing or coherent perfect absorption of pairs of oppositely polarized modes. Strong spectral singularities are characterized by zero diagonal block of the transfer matrix. We show that in odd-$PT$-symmetric systems any spectral singularity is self-dual, i.e., lasing and coherent perfect absorption occur simultaneously. Detailed analysis is performed for a case example of a $PT$-symmetric coupler consisting of two waveguides, one with localized gain and another with localized absorption, which are coupled by a localized antisymmetric medium. For this coupler, we discuss weak self-dual spectral singularities and their splitting into complex-conjugate eigenvalues which represent bound states characterized by propagation constants with real parts lying in the continuum. A rather counterintuitive restoration of the unbroken odd-$PT$-symmetric phase subject to the increase of the gain-and-loss strength is also revealed. The comparison between odd- and even-$PT-$symmetric couplers, the latter characterized by $(PT)^2=1$, is also presented.