Tunneling through inhomogeneous delta barriers as diffraction and scattering

We present a general theory of the quantum-mechanical tunneling through an inhomogeneous planar delta barrier. The delta barrier means that the potential energy is proportional to the Dirac delta function in the direction perpendicular to the plane of the barrier. The inhomogeneity of the delta barrier means that the delta function is multiplied by a non-negative function, different from a constant, of lateral coordinates. We assume that this function, being the barrier strength, may be arbitrary. To exemplify our theory, we consider delta barriers that are made inhomogeneous by embedding equal circular windows in the barrier plane, assuming that the barriers are homogeneous both inside and outside the windows. (The value of the barrier strength is taken higher by the side of the windows than inside the windows.) With the inhomogeneous delta barriers of this kind, we show how the tunneling theory is related to the theory of diffraction and scattering. Although our general solution of the problem is new in the context of the tunneling theory, it is essentially based on a method which was used by Kirchhoff in the 19th century in the theory of waves.