Green function singularities

Furthermore, if d is much smaller than the coherence length = yftP/A, these Green functions take the BCS form, with the phase-dependent proximity gap A,p = A cos(c/)/2) [18]. This results in the BCS-like singularity at the gap edge in the density of states (DOS) of the normal wire and suppression of the DOS at E < A,/,. Within such model, the problem of current statistics in the Andreev interferometer reduces to the calculation of the CGF for an NS junction with the effective order parameter A,p in the superconducting reservoir. 

If the matrix C is not singular, which requires the number of basis functions to match the number of solid harmonics used to expand the Green function, a local r-matrix is defined by t = —SC l. The consistency condition expressed above in terms of C and S matrices then reduces to the simple matrix expression... 

G is the reduced Green function of the Schrodinger equation and B = (Us)-Action of the operator O2 on the wave function can be checked not to produce functions more singular than G2 or c2. Therefore, in contrast to the second iteration of the original perturbation, Eq.(12), that of the operator 02 delivers a result which is finite in three dimensions. 

Two different procedures were used for the removal of the reference state singularities and the derivation of RSC. As it was shown in [12], the Coulomb-Coulomb contribution to RSC is absent. The RSC were first introduced in QED in [13] in the frames of the Green function formalism in the Feynman gauge [13,14]. The expressions for RSC are much simplified in the equal energies case (Ea = Eb), i.e. for the ground state of the two-electron ion. The explicit expression for the equal energies correction was derived in [15]... 

Expression (231) and, consequently, Eq(229) are divergent. This divergency follows from the singularity of the Green function... 

It is noticed that the Green s function A (x) has one singularity when Xy = 0, but it can be eliminated by an integration over the element around the point Xy=0. A computation experiment shows that Eq (30) may result in a significant numerical error when coarser grids are employed. 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

As follows from Eq. (12), the Green s function does not have a singular part, but a self-energy operator, , does have it [42]... 

Following ideas of the extended Born approximation (Habashy et al., 1993), we recall that the Green s function G (r r o ) exhibits either singularity or a peak at the point where = r. Therefore, one can expect that the dominant contribution to the integral G [As Ap ] in equation (14.43) is from some neighborhood of the point Tj — r. In fact, we can expand A (r,cu) into a Taylor series about r = r ... 

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