Logarithmic matrix function

To simplify our discussion of log A, we restrict the domain of the logarithmic matrix function to nonsingular matrices A that can be diagonalized... 

Fig. 3.1. The relationship between the exponential and logarithmic matrix functions. The domains for one-to-one mappings in the neighbouihoods of 0 and 1 are indicated by circles.

The logarithmic matrix function log A is thus the true inverse function of the exponential expA in the neighbourhood of 1, mapping matrices in the neighbourhood of 1 onto matrices in the neighbourhood of 0. 

The following properties of the logarithmic matrix function are proved in Exercise 3.6 ... 

Prove the following relations for the logarithmic matrix function ... 

Since A, B and AB belong to the domain where the exponential matrix function is the inverse of the logarithmic matrix function, we obtain... 

In 1973 Andersen and Woolley [1.25] extended the LCMTO method to molecular calculations. At the end of their paper they introduced that choice of MTO tail, i.e. proportional to J = 9i/j/9E, which in a natural fashion ensured orthogonality to the core states and at the same time led to an accurate and elegant formulation of linear methods. The resulting, technique was immediately developed in a paper by Andersen [1.26] which, in a condensed form, contains most of what one need know about the simple concepts of linear band theory. Thus, we find here the KKR equation within the atomic-sphere approximation at this stage is called ASM the LCMTO secular matrix, latter called the LMTO matrix the energy-independent structure constants and the canonical bands and the Laurent expansion of the logarithmic-derivative function and the corresponding potential parameters. 

In Exercise 3.6, the proof given for (3.4.14) is valid only for matrices belonging to the domain where the logarithmic and exponential matrix functions are inverse functions. Finally, (3.4.15) - which follows from (3.4.12) and (3.4.13) - shows that the logarithmic function maps unitary matrices onto anti-Hermitian matrices as expected from the fact that flie exponential function maps anti-Hermitian matrices onto unitary matrices. 

Fig. 31.14. Performance of three computer algorithms for eigenvalue decomposition as a function of the dimension of the input matrix. The horizontal and vertical scales are scaled logarithmically. Execution time is proportional to a power 2.6 of the dimension.

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

Note that the numerical factor before the leading logarithm here is simply the product of the respective numerical factors in the correction to the wave function in (3.65), the low-frequency asymptote of the one-loop polarization — 1/(157t) in (2.5), the factor 47r(Za) in (3.62), and factor 2 which reflects that both wave functions in the matrix element in (3.62) have to be corrected. 

It is evident that this potential leads to a logarithm squared contribution of order a (Za) after substitution in (3.71). One may obtain one more logarithm from the continuous spectrum contribution in (3.71). Due to locality of the potential, matrix elements reduce to the products of the values of the respective wave functions at the origin and the potentials in (3.72). The value of the continuous spectrum Coulomb wave function at the origin is well known (see, e.g., [94]), and... 

We applied the generating functional approach to the periodic Anderson model. Calculation of the electron GFs gdd, 9ds, 9sd and gss reduces to calculation of only the d-electron GF. For this, an exact matrix equation was derived with the variational derivatives. Iterations with respect to the effective matrix element Aij(to) allow to construct a perturbation theory near the atomic limit. Along with the self-energy, the terminal part of the GF Q is very important. The first order correction for it describes the interaction of d-electrons with spin fluctuations. In the paramagnetic phase this term contains a logarithmic singularity near the Fermi-level and thus produces a Kondo-like resonance peak in the d-electron density of states. The spin susceptibility of d-electrons... 

The theory of the / -matrix was developed in nuclear physics. As usually presented, the theory makes use of a Green function to relate value and slope of the radial channel orbitals at r, expanding these functions for r < r as linear combinations of basis functions that satisfy fixed boundary conditions at r. The true logarithmic derivative (or reciprocal of the / -matrix in multichannel formalism)... 

The theory of the / -matrix can be understood most clearly in a variational formulation. The essential derivation for a single channel was given by Kohn [202], as a variational principle for the radial logarithmic derivative. If h is the radial Hamiltonian operator, the Schrodinger variational functional is... 

This is the reciprocal of the logarithmic derivative of the wave function Xdiq) in the dissociation channel, for q = qd. At given total energy E these R-matrix elements are matched to external scattering wave functions by linear equations that determine the full scattering matrix for all direct and inverse processes involving nuclear motion and vibrational excitation. Because the vibronic R-matrix is Hermitian by construction (real and symmetric by appropriate choice of basis functions), the vibronic 5-matrix is unitary. 

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