Two-photon ionization of atomic hydrogen with an excess photon is revisited. The non-relativistic dipole approximation and Coulomb Green function (CGF) formalism are applied. Using the CGF Sturmian expansion straightforwardly, one gets the radial transition amplitude in the form of an infinite sum over Gauss hypergeometric functions which are polynomials. It is convergent if all intermediate states are in the discrete spectrum. In the case of two-photon ionization with an excess photon, when photoionization is also possible, intermediate states are in the continuum. We performed the explicit summation over intermediate states and got a simple general expression for the radial transition amplitude in the form of a finite sum over Appell hypergeometric functions, which are not polynomials. An Appell function may be expressed as an infinite sum over Gauss functions. In the case of ionization by an excess photon, Gauss functions are transformed to give a convergent radial transition amplitude for the whole region. The generalized cross sections for two-photon above-threshold ionization of atomic hydrogen in the ground state and excited states calculated by us agree very well with results of previous calculations. Generalized cross sections for two-photon ionization of positronium in the ground state are obtained by scaling those for atomic hydrogen.
We study the Green's function of the $\ensuremath{\nu}=1/2$ Chern-Simons system in the temporal (Weyl) gauge. We derive the Chern-Simons path integral in the temporal gauge. In order to do this, we gauge transform the path integral in the Coulomb gauge which represents the partition function of the correct normal ordered Chern-Simons Hamiltonian. We calculate the self-energy of this path integral in the random-phase approximation (RPA) for temperature
T=0. This self-energy does not have the divergence with the logarithm of the area, which is known to imply the vanishing of the exact Green's function in the Coulomb gauge for an infinite area. By Chern-Simons retransforming the path integral representing the Green's function in the temporal gauge we calculate explicitly the exact Green's function by neglecting the interaction between the electrons, getting a finite value. Furthermore, we give arguments that the Green's function of the interacting system is also finite. The nonvanishing of the Green's function for infinite area is due to a dynamical creation of the phase factors linking the created and annihilated particles with the particles in the ground state. The absence of these phase factors is the reason for the vanishing of the Green's function in the Coulomb gauge.
The Schrodinger-Coulomb Green function is discussed as a tool for evaluating terms in bound-state perturbation theory. As an example the energy shift of the ground state of hydrogen in a uniform electric field is considered. The Green function is used in an evaluation of the second-order perturbation theory expression for this energy shift. For comparison a direct numerical approach to the evaluation of this energy shift is also described. The Green function approach is simpler and more elegant, and leads to an exact result.