Resolvent operator

The operator T i j is called the transition operator from the layer j to the layer n + l, while the operator o refers to the resolving operator. 

If the condition of uniform stability is satisfied, then estimate (19) holds true for the resolving operator T j- Therefore, Theorem 1 asserts that estimate (20) is valid for a solution of problem (4). This type of situation is covered by the following results. 

Sufficiency. The stability with respect to the initial data means the boundedness of the resolving operator... 

From such reasoning it seems clear that the condition of termination is ensured if g < e, thereby reducing the question of convergence of the iterations to the norm estimation of the resolving operator T . 

Here T , is the resolving operator being a polynomial of degree n with respect to the operator A ... 

S - transition operator from one layer to another T - resolving operator ... 

The wavefunction corrections can be obtained similarly through a resolvent operator technique which will be discussed below. The n-th wavefunction correction for the i-th state of the perturbed system can be written in the same marmer as it is customary when developing some scalar perturbation theory scheme by means of a linear combination of the unperturbed state wavefunctions, excluding the i-th unperturbed state. That is ... 

However, these formal expressions are of no great help until we know how to operate with the exponential operator, which is very complicated because it involves the full iV-body problem with the interactions between the particles. In order to circumvent this difficulty, we shall use a resolvant technique 89 we define a resolvant operator (L — 2)-1, function of the complex variable z, and write ... 

Before finishing our formal manipulations, we still have to express the distribution functions (30) and (31) in terms of the resolvant operator it is not difficult to show that (see Eq. (32))... 

A similar problem was discussed in Section IV-C, where the motion of a single ion in the solvent was considered. We shall briefly indicate in Appendix A2 how this calculation may be generalized to the case where the heavy particle has a nonvanishing wave number. The resolvant operator is decomposed into irreducible diagonal fragments, which are then expanded in powers of yx the result is ... 

Let us introduce the resolvent operator, a function of the complex variable z ... 

The propagator can be obtained as a Fourier integral of the resolvent operator... 

The above resolvent operator Hi E) refers to the operator H° including only the Coulomb interaction between the subsystems. Its poles ) are those eigenvalues of H° which differ from those in the subspace Im P by transfers of one electron between the M- and R-systems. We denote these states as p — /. ) or /. — p) with respect to the direction of the transfers. The energies in the expression in eq. (1.242) are defined by the ionization potentials Ip and electron affinities Afl, Ap of the subsystems ... 

In addition to Eq. (11), it is also useful to introduce the Green function R(u) as the matrix elements of an alternative resolvent operator associated with U ... 

No matrix inversion is encountered in Eq. (230). Stated equivalently, the matrix ul — U associated with the resolvent operator R(n) from Eq. (48) is inverted iteratively through its corresponding LCF. The LCF as a versatile convergence accelerator can yield the frequency spectrum (230) with... 

The resolvent operator as a function of E has poles at the eigenvalues of Ho- Strictly speaking, for the expression in Eq. (2) to be meaningful, the singular component of the resolvent must be removed, but we choose not to clutter the notation with formalism that is unnecessary for our purposes. We note that Eq. (1) can be derived from Eq. (2) if we insert a resolution of the identity (and removing the ground state) on each side of the resolvent,... 

As soon as the eigenfunctions and the eigenvalues are known, it is possible to find the resolvent operator (see, for example, Ref.39 ). Using this operator, the solution of Eq. (5.22) may be readily obtained as... 

A. L. Ford and J. C. Browne. Direct-resolvent-operator computations on the hydrogen-molecule, dynamic polarizability, Rayleigh, Raman scattering. Phys. Rev. A, 7 418-426 (1973). 

The detailed emulsion characterization methods discussed herein can be used to help resolve operational upsets only if a base line of data exists for normal operation. In fact, without a thorough characterization of the normal emulsion properties such as size distribution and mineral and organic composition, the techniques for detailed characterization may actually hinder the understanding and ultimate solution of a particular processing problem by introducing extraneous information. When a base line of data exists, detailed information on the size distribution and the relationship between the dispersed, continuous, and solid phases is invaluable. 

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