Summary: Non-self-adjoint Schrödinger operators \(A_{\mathfrak{T}}\) which correspond to non-symmetric zero-range potentials are investigated. We show that various properties of \(A_{\mathfrak{T}}\) (eigenvalues, exceptional points, spectral singularities and the property of similarity to a self-adjoint operator) are completely determined by poles of the corresponding \(S\)-matrix.