Wave systems with direct processes and localized losses or gains: the non-unitary Poisson kernel

We study the scattering of waves in systems with losses or gains simulated by imaginary potentials. This is done for a complex delta potential that corresponds to a spatially localized absorption or amplification. In the Argand plane the scattering matrix moves on a circle $C$ centered on the real axis, but not at the origin, that is tangent to the unit circle. From the numerical simulations it is concluded that the distribution of the scattering matrix, when measured from the center of the circle $C$, agrees with the non-unitary Poisson kernel. This result is also obtained analytically by extending the analyticity condition, of unitary scattering matrices, to the non-unitary ones. We use this non-unitary Poisson kernel to obtain the distribution of non-unitary scattering matrices when measured from the origin of the Argand plane. The obtained marginal distributions have an excellent agreement with the numerical results.