Regular expansion

As shown by Chang, Pelissier and Durand (CPD) [41] a regular expansion, however, can be deduced by isolating the Coulomb singularity by infinite summations. Let us rewrite the equation (38), when z — 0... 

The solution (2.3) is known as a regular perturbation expansion, Xio(f) is the solution of the original problem (2.1), and the higher-order terms xi i(t),... are determined successively by substituting the regular expansion (2.3) into the original differential equation (2.1) (Haberman 1998). 

A very well-studied technique to arrive at regular expansions was developed in the mid 1980s (Chang et al. 1986 Heully et al. 1986). The essential point is to rewrite the prefactor of cap in (3.4) as... 

We now seek a solution of (9 7) and (9-8) for small values of the Peclet number, Pe , by using the matched asymptotic expansion procedure that was detailed for uniform flow past a sphere in Section C. Although the reader may not immediately see that the derivation of an asymptotic solution for this new problem necessitates use of the matched asymptotic expansion technique, an attempt to develop a regular expansion for 9 for Pe 1 leads to a Whitehead-type paradox similar to that encountered for the uniform-flow problem. 

For 0 < Ifi l < 1 and as Pe -> 00, the concentration is uniform in the region with open streamlines and equal to its value at infinity the concentration distribution in the region filled with closed streamlines can be represented as a regular expansion in powers of the inverse Peclet number ... 

It is easy to see that 3 p(S) has a regular expansion in powers of z. Thus, the anomaly at dimension d = 4 — 2jp simply results in an additional renormalization factor which is singular with respect to z and of the form... 

The renormalization factors cancel out, and for the physical quantities under consideration, perfectly regular expansions in powers of z,..., zN are obtained. They are given by... 

In the mid-eighties another method to eliminate the small component has been developed in order to arrive at regular expansions for the Hamiltonian [13,26]. These regular approximations are based on the general theory of effective Hamiltonians [27,28], where the full problem under consideration is projected onto a smaller, suitably chosen model space with an effective Hamiltonian, which comprises all desired properties of the problem sufficiently well. In the case of the Dirac Hamiltonian the basic idea, being as simple as ingenious, is to rewrite the expression for w in the form... 

Liquid-solid fluidised systems are generally characterised by the regular expansion of the bed which takes place as the liquid velocity increases from the minimum fluidisation velocity to a value approaching the terminal falling velocity of the particles. The general form of relation between velocity and bed voidage is found to be similar for both Newtonian and inelastic power-law liquids. For fluidisation of uniform spheres by Newtonian liquids, equation (5.21), introduced earlier to represent hindered settling data, is equally applicable ... 

For a Coulomb potential the cut-off function / varies from 0 to 1 as illustrated in Fig. 7. The iterative solution of (103) provides regular terms that finally lead to a regular expansion of the Bloch and des Cloizeaux effective Hamiltonians. The first terms are ... 

If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters The expansion we consider in this chapter has roots in the work by Chang, Pelissier, and Durand (1986) and HeuUy et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term regular approximation because of the properties of the expansion. 

The first-order approximations discussed in the next section, although historically derived from non-regular perturbation series, yield valid first-order expectation values, and thus their value should agree with the first-order expectation value of any regular expansion in 1/c, since the latter is unique as far as the series is regular. 

A more promising route is to look for a perturbation parameter which permits a regular expansion for all values of the momentum. Chang, Pelissier, and Durand (CPD) have found a corresponding perturbation series, and the Amsterdam group " has developed it further. It makes use of... 

As a first step towards construction of a regularized SAPT theory that includes both Vp and Vt, let us note that if V p and Sp are known, the remaining part of the interaction energy can be recovered by means of a perturbation expansion in powers of Vt. Since this expansion does not influence the, already correct, asymptotics of 8, we can now employ a method that is convergent despite the presence of the Pauli-forbidden continuum, i.e. the EL-HAV, AM, or JK theory. Unlike the case of non-regularized expansions, there is no asymptotics-related reason to expect that the JK method will perform better than the other two. If we choose the ELHAV theory, the successive corrections to the energy and the wave function, referred to as the regularized ELHAV (R-ELHAV) corrections, are obtained from the formulae [40]... 

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