Singular term

As it is seen from the derivation, only the second, non-singular term... 

This problem is simplified for d = 1,3 since in these two particular cases the singular term d becomes zero and thus there is no limitations for... 

In its turn, for d = 2 the singular term (d — l)(d — 3)/(4tj2) prevents use of the VKB in a whole interval of r) values. However, a special analysis carried out in [30] has demonstrated that this singularity does not affect the final result (6.2.24). So far, the relation (6.2.24) holds for the space dimensions... 

For d 4 singular term (d — l)(d — 3)/(4 j2) does not allow to find the solution at r/ = 0. It has simple interpretation in systems with so large space dimensionalities no variable rj = r/fo exists there. Similar to d = 3 in the linear approximation, for d 4 we can find the stationary solutions, Y (r, oo) = y0(r). For them the reaction rate K(oo) = Kq — const and the classical asymptotics n(t) oc Ya, ao = 1 hold. Therefore, for a set of kinetic equations derived in the superposition approximation the critical space dimension could be established for the diffusion-controlled reactions. 

Separating now the singular terms of equations (p (r) = p,(r)) containing -functions, we arrive at the set of equations for the joint densities ... 

Therefore the singular terms with p(r) enter equations for the joint densities of similar particles only. We show below how these terms modify the relevant boundary conditions at the coordinate origin. 

Strictly speaking, equations for the joint densities of similar particles have to be solved with the boundary condition (5.1.40) imposed at the coordinate origin. However, the singular terms S(r — r ) with r — 0 have to modify it. To illustrate this point, let us consider the generalized diffusion equation with the singular term... 

The derivative is replaced here by the difference.) Therefore the final kinetic equation with the boundary condition modified due to the singular term is quite similar to (5.1.40) discussed above. Therefore the singular terms in equations for the joint densities (8.2.15) and (8.2.16) for similar particles could be also omitted ... 

That is, the singular terms containing p(r) enter equation (8.3.7) for pop, (the joint density of similar particles) only. As earlier, these terms are used to modify the boundary conditions at the coordinate origin. 

As it was shown in Section 8.2, the singular term in (8.3.23) could be omitted due to the transformation of the boundary condition at coordinate origin for the correlation function As(r, t),... 

Equation (5.66) contains the singular term —2 (y2(iv)/iv). This is singular for w = 0 since it asks for the division of 2/2(0) = 0 by zero there. To account for this in MATLAB and to avoid dividing by zero, we define the SingularTerm vector used in MATLAB s BVP solver as the coefficient matrix... 

Further complications of the DE (5.71) arise due to the boundary conditions which require us to define a singular term matrix at w = 0, see p. 315 for more details. 

The main equation for the d-electron GF in PAM coincides with the equation for the Hubbard model if the hopping matrix elements t, ) in the Hubbard model are replaced by the effective ones Athat are V2 and depend on frequency. By iteration of this equation with respect to Aij(u>) one can construct a perturbation theory near the atomic limit. A singular term in the expansions, describing the interaction of d-electrons with spin fluctuations, was found. This term leads to a resonance peak near the Fermi-level with a width of the order of the Kondo temperature. The dynamical spin susceptibility in the paramagnetic phase in the hydrodynamic limit was also calculated. 

Remark A.l. Condition (i) of Theorem A.l essentially means that the corresponding DAE system in Equation (2.45) has an index of two, which directly fixes the dimensions of the fast and slow variables to p and n—p, respectively. Condition (ii) of the theorem ensures that the (n — p)-dimensional slow C,-subsystem can be made independent of the singular term /e, thereby yielding the system in Equation (A.13) in the standard singularly perturbed form. While condition (n) is trivially satisfied for all linear systems and for nonlinear systems with pi = 1, it is not satisfied in general for nonlinear systems with p> > 1. 

It yields the finite physical energy shift A E in the limit A -+ oo. Singular terms arise from the high-energy region of the integration over E. They can be isolated in the first few terms in the expansion of the electron propagator... 

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