Considerations on describing non-singlet spin states in variational second order density matrix methods

2012 ◽  
Vol 136 (1) ◽  
pp. 014110 ◽  
Author(s):  
Helen van Aggelen ◽  
Brecht Verstichel ◽  
Patrick Bultinck ◽  
Dimitri Van Neck ◽  
Paul W. Ayers
Keyword(s):  
Density Matrix ◽  
Second Order ◽  
Spin States ◽  
Matrix Methods

Transformations between second order linear differential equations and a reinterpretation of recurrence relations between Bessel functions with algebraic coefficients

1977 ◽  
Vol 79 (1-2) ◽  
pp. 87-105 ◽  
Author(s):  
John Heading
Keyword(s):  
Differential Equations ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Asymptotic Methods ◽  
Second Order ◽  
Schwarzschild Black Hole ◽  
Matrix Methods ◽  
Large Numbers ◽  
Method Of Solution ◽  
Linear Differential

SynopsisA scheme devised by Chandrasekhar for investigating the transformations between various differential equations of the second order governing perturbations of the Schwarzschild black hole demands further investigation. The transformation between two differential equations in normal form is considered, and a wide survey of the properties of the transformation is given. It is shown how Chandrasekhar's equations fit into the scheme, after which some examples with particular properties are considered. A detailed investigation of Bessel's equation is undertaken using various devices, in particular by employing asymptotic methods for products of Bessel functions, and employing matrix methods for dealing with large numbers of matrix equations which necessitates an interesting method of solution, the results being reinterpretations of the standard recurrence relations for Bessel functions.

GORKOV'S ANSATZ IN SUPERCONDUCTIVITY THEORY

1965 ◽  
Vol 43 (12) ◽  
pp. 2142-2149 ◽  
Author(s):  
A. J. Coleman ◽  
S. Pruski
Keyword(s):  
Density Matrix ◽  
Green’S Function ◽  
Green's Function ◽  
Order Parameter ◽  
Green’S Functions ◽  
Second Order ◽  
Bcs Theory ◽  
Initial Value ◽  
Ginzburg Landau ◽  
Superconductivity Theory

By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.

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