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Register a new userA generating integral for the matrix elements of the Coulomb Greens function with the Coulomb wave functions
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We analytically evaluate the generating integral $K_{nl}(beta,beta) = int_{0}^{infty}int_{0}^{infty} e^{-beta r - beta r}G_{nl} r^{q} r^{q} dr dr$ and integral moments $J_{nl}(beta, r) = int_{0}^{infty} dr G_{nl}(r,r) r^{q} e^{-beta r}$ for the reduced Coulomb Greens function $G_{nl}(r,r)$ for all values of the parameters $q$ and $q$, when the integrals are convergent. These results can be used in second-order perturbation theory to analytically obtain the complete energy spectra and local physical characteristics such as electronic densities of multi-electron atoms or ions.
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In this paper the fixed-energy amplitude (Greens function) of the relativistic Coulomb system is solved by Duru-Kleinert (DK) method. In the course of the calculations we observe an equivalence between the relativistic Coulomb system and a radial oscillator.
In a previous work [Andrade textit{et al.}, Phys. Rep. textbf{647}, 1 (2016)], it was shown that the exact Greens function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effects are taken into account through the reflection and transmission scattering amplitudes. To deal with general graphs, two simplifying procedures were developed: regrouping of paths into families of paths and the separation of a large graph into subgraphs. However, for less symmetrical graphs with complicated topologies as, for instance, random graphs, it can become cumbersome to choose the subgraphs and the families of paths. In this work, an even more general procedure to construct the energy domain GF for a quantum graph based on its adjacency matrix is presented. This new construction allows us to obtain the secular determinant, unraveling a unitary equivalence between the scattering Schrodinger approach and the Greens function approach. It also enables us to write a trace formula based on the Greens function approach. The present construction has the advantage that it can be applied directly for any graph, going from regular to random topologies.
The Dirac equation in $mathbb{R}^{1,3}$ with potential Z/r is a relativistic field equation modeling the hydrogen atom. We analyze the singularity structure of the propagator for this equation, showing that the singularities of the Schwartz kernel of the propagator are along an expanding spherical wave away from rays that miss the potential singularity at the origin, but also may include an additional spherical wave of diffracted singularities emanating from the origin. This diffracted wavefront is 1-0 derivatives smoother than the main singularities and is a conormal singularity.
A matrix basis formulation is introduced to represent the 3 x 3 dyadic Greens functions in the frequency domain for the Maxwells equations and the elastic wave equation in layered media. The formulation can be used to decompose the Maxwells Greens functions into independent TE and TM components, each satisfying a Helmholtz equation, and decompose the elastic wave Greens function into the S-wave and the P-wave components. In addition, a derived vector basis formulation is applied to the case for acoustic wave sources from a non-viscous fluid layer.
In this paper we study the time dependent Schrodinger equation with all possible self-adjoint singular interactions located at the origin, which include the $delta$ and $delta$-potentials as well as boundary conditions of Dirichlet, Neumann, and Robin type as particular cases. We derive an explicit representation of the time dependent Greens function and give a mathematical rigorous meaning to the corresponding integral for holomorphic initial conditions, using Fresnel integrals. Superoscillatory functions appear in the context of weak measurements in quantum mechanics and are naturally treated as holomorphic entire functions. As an application of the Greens function we study the stability and oscillatory properties of the solution of the Schrodinger equation subject to a generalized point interaction when the initial datum is a superoscillatory function.
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