Memory function methods

In Table II we report the same illustrative example following the Gordon procedure. The convenience of the memory function method is apparent. Furthermore, the Gordon method fails unless 0, a restriction which is overcome in the memory function method. Another quite interesting P-algorithm has been provided by H ggi et al., and we refer to the original papers for illustration and discussion of stability aspects. 

MEMORY FUNCTION METHODS IN SOLID STATE PHYSICS... 

V. Application of Memory Function Methods in Periodic and Aperiodic Solids.163... 

In this section we briefly survey a number of physical problems whose Hamiltonians can be conveniently described in a local basis. Hamiltonians of this kind allow a reasonably simple calculation of moments and are thus natural candidates for the memory function methods we are going to describe. 

MEMORY FUNCTION METHODS IN SOLID STATE PHYSICS bor interactions only can be written diagrammatically as ... 

Once a tridiagonal representation of M is obtained, one can use standard methods to diagonalize the tridiagonal matrix and obtain the eigenvalues (in this particular aspect, the Lanczos algorithm, as commonly used in the literature, differs from the other memory function methods where the Green s... 

V. APPUCATIONS OF MEMORY FUNCTION METHODS IN PERIODIC AND APERIODIC SOLIDS... 

In this section we give a few illustrative applications of memory function methods in solids. The subject, even when confined to electronic structure, is too wide for a thorough account, and by necessity a selection has to be made. 

We remark that memory function methods have been extensively applied to perfect crystals, where they are alternative tools to the traditional band structure methods. In the presence of translational symmetry and long-range order, however, the use of the memory function techniques is not essential but rather a matter of convenience (or taste) in a number of situations. We thus focus on the study of the electronic structure in systems that are aperiodic because of impurities our unorthodox way of looking at defects is to consider them as a source of a given frozen-in disorder in an otherwise perfect crystal lattice. The more general case of the presence of stochastic disorder will be discussed in Section VI. 

The problems associated with defects are quite numerous, and a wealth of theoretical techniques have been devised for their solution. We shall confine our attention to the Green s function formalism because of both the level of accuracy and sophistication of recent applications.and the promise that memory function methods hold in this field. 

MEMORY FUNCTION METHODS IN SOUD STATE PHYSICS where the matrix elements p in the localized basis are defined as... 

Summing up, we see that the traditional approach to impurity problems within the Green s-function formalism exploits the basic idea of splitting the problem into a perfect crystal described by the operator and a perturbation described by the operator U. The matrix elements of < are then calculated, usually by direct diagonalization of or by means of the recursion method. Following this traditional line of attack, one does not fully exploit the power of the memory function methods. They appear at most as an auxiliary (but not really essential) tool used to calculate the matrix elements of... 

A less orthodox line of attack, as yet not explored to its full potential, applies from the beginning the recursion method to the solid-plus-impurity system. The direct use of memory function methods to the perturbed solid is no more difficult than for the perfect solid, with the advantage of overcoming the traditional separation of the actual Hamiltonian into a perfect part and a perturbed part. In fact, such a separation, to make any practical sense, requires that the perturbed part be localized in real space, a restriction hardly met when treating impurities with a coulombic tail. 

In recent years there has been an explosion of interest in the electron properties of disordered lattices. The more common line of approach to this kind of problem is to study the mean resolvent of the random medium, and the memory function methods can be of remarkable help for this purpose. Otherwise one can investigate by the memory function methods (basically the recursion method) a number of judiciously selected configurations this line of approach is particularly promising because it allows one to overcome some of the limitations inherent in the mean field theories. In this section we de-... 

The technique of combining CPA concepts with a continued fraction analysis of the Green s function should also be invaluable in connection with the study of disorder with augmented space techniques. Progress is expected when memory function methods are systematically used in the mentioned areas of investigation. 

TV. Memory Function Methods in Solid State PIqrsics 133... 

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