Eigenvalue equation poles

It has been shown explicitly by Casida" that the poles of the polarizability (i.e., the excitation energies) occur at the eigenvalues w, of the following eigenvalue equation... 

The simplest truncation of the eigenvalue equation (2) for the excitation energies is to ignore all coupling between poles, except that between a singlet-triplet pair. This is equivalent to setting (gl/nxcl ) to zero, for q q. (We have dropped the spin-index on these contributions, since we deal only with closed shell systems). Then the eigenvalue problem reduces to a simple 2x2 problem, with solutions... 

The above decomposition complements our intuitive description of leaky modes in Chapter 24, and shows formally that the fields of leaky modes have the same analyticalforms as bound-modefields. Each leaky mode is associated with a transverse resonance in the fiber cross-section, specified by the eigenvalue equation Wi(U,Q) = 0. Leaky modes only contribute to the radiationfield within an angular sector of the (r,z)-plane of Fig. 26-2. Consider the qith leaky-mode pole with coordinates (steepest descent path sd and the path p in Fig. 26-1 (b). In this case (J, and f/, satisfy the inequalities in Eq. (26-10). As r and z vary, the angle 6 defined by Eq. (26-10) will vary and the steepest descent path defined by Eq. (26-9) will be continuously modified. Clearly, there is a maximum angle 0, for which Eq. (26-10) is only just satisfied, when the leaky-mode pole lies on the steepest descent path. In this case. 

The roots of equation (8.95) are the open-loop poles or eigenvalues. For the closed-loop system described by equation (8.94), the characteristic equation is... 

The roots of equation (8.96) are the closed-loop poles or eigenvalues. 

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

We next return to our assertion that we can choose all our closed-loop poles, or in terms of eigenvalues, ib K- This desired closed-loop characteristic equation is... 

This Exercise demonstrates that some care is needed in utilizing (4.3) for finding n and x because pRtm(X) may have a singularity X = 0. Let us suppose in particular that the M-equation has a discrete eigenvalue spectrum, as in V.7. The lowest eigenvalue is zero, with eigenfunction psn = pn m(oo). Then the Laplace transform has a pole,... 

The complex eigenvalues of F are also poles of t, and therefore they are solutions of the equations... 

These poles are the roots of the denominator of all the elements in equation (46b). It is important to note that both the eigenvalue characteristic polynomial and the... 

With appropriate dimensional scalings, theD — 1 limit of Schrodinger equations for coulombic systems provides important information about the dimension dependence of energy eigenvalues. The energy typically has a second-order pole at D = 1, the residue of which can often be exactly determined. We demonstrate this with some simple examples and then review a systematic procedure for characterizing a class of dimensional singularities found in coulombic problems. 

The poles of the linear susceptibility, x(fr fi)), are the excitation frequencies of the true system. In order to extract these frequencies Casida used ancient RPA technology to produce equations in which these poles of % are found as the solution to an eigenvalue problem. In order to see this, first do an expansion of the density change in the basis of KS transitions. We write hn (ri) as... 

Excitation energies are thus computed as poles of the dynamic polarizability, that is, as the values of co leading to zero eigenvalues on the left-hand side of the matrix of Eq. 1.12. In the framework of the above equations, an efficient fast iterative solution for the lowest eigenvalue/excitation energies can be attained [62]. Oscillator strengths can also be obtained by the eigenvectors of Eq. 1.12, as explained by Casida [17]. 

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