Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

International Journal of Modern Physics B

SOME REMARKS ON THE EQUATION OF MOTION METHOD FOR THE GREEN’S FUNCTION CALCULATIONS

Ryszard Taranko

Published Sep 1, 1991• 0 citations
Publisher

Abstract

The retarded, double-time Green’s Functions are studied using different modifications of the projection operator technique. It is shown that the results of all these approaches can be written in the form of the finite or infinite continued fractions. Moreover, they lead to the same expansions for the Green’s Functions when terminated at the same level.

DOI
10.1142/s0217979291000985

Related papers

Classical Green's Functions for the Ideal Gas

Michael J. Smith

We use the equation-of-motion technique to establish a chain of equations for retarded, double-time Green's functions of an ideal gas. Exact termination of this chain allows reproduction of the usual results obtained by other methods.

Physical Review Letters•Jun 22, 1970•3 citations

Equation-of-motion series expansion of double-time Green's functions

Ning-Hua Tong

Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter $\ensuremath{\lambda}$ in the Hamiltonian $H={H}_{0}+\ensuremath{\lambda}{H}_{1}$. Here ${H}_{0}$ is the exactly solvable part and ${H}_{1}$ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order ${\ensuremath{\lambda}}^{2}$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.

arXiv:1507.06407•Physical Review B•Oct 26, 2015•8 citations

Convergence of continued-fraction representation for the Green’s function in the Hubbard model

Chang Ho Choi, Yong Jai Cho, Hyo Suk Yi +1 more

Using the operator algebra technique with a set of projection operators generated from the primary operator defined by Wilson's group [S. K. Kim and R. S. Wilson, Phys. Rev. A 7, 1396 (1973)], we set up a continued-fraction representation for the single-particle Green's function in the Hubbard model. We find that, in a simple one-dimensional system with five lattice points, at least one cutoff stage appears in the representation.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Sep 1, 1995•0 citations

A continued fraction representation of the mass operator

Dipak K. Saraswati

We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.

Canadian Journal of Physics•Jan 1, 1976•1 citation

Equation of Motion Theory

Henrik Bruus, Karsten Flensberg

Abstract In the previous chapters, we saw how various physical observables can be expressed in terms of retarded Green’s functions. In many cases we need to calculate the time dependence of these functions. There are several ways of attacking this problem, one of which is the equation of motion technique. The basic idea of this method is to generate a series of coupled differential equations by differentiating the function at hand a number of times. If these equations close, the problem is in principle solvable; and if not, one needs to invoke physical arguments to truncate the set of equations in a reasonable fashion. For example, one can neglect certain correlations. We shall study examples of both situations in this chapter.

•Sep 2, 2004•0 citations

Properties of advanced coupled-cluster Green's function

Bo Peng, Karol Kowalski

We present an extension of the analysis previously applied to the retarded part of the coupled cluster (CC) Green's function to its advanced part. In analogy to our earlier studies for the retarded part, we demonstrate that the advanced CC Green's function is expressed in terms of connected diagrams only, which is a direct consequence of algebraic form of equations satisfied by CC amplitudes. We also demonstrate that ω-derivatives of the advanced CC Green's function can be calculated analytically and can be expressed in terms of connected diagrams only. We analyse the structure of connected diagrams and the role of intermediate operators which satisfy electron affinity equation-of-motion CC-type conditions.

Molecular Physics•Jul 25, 2017•16 citations
The retarded, double-time Green’s Functions are studied using different modifications of the projection operator technique. It is shown that the results of all these approaches can be written in the form of the finite or infinite continued fractions. Moreover, they lead to the same expansions for the Green’s Functions when terminated at the same level.
Based on the Green's function (GF) equation-of-motion formalism, we develop a method to expand the double-time Green's function into Taylor series of the parameter $\ensuremath{\lambda}$ in the Hamiltonian $H={H}_{0}+\ensuremath{\lambda}{H}_{1}$. Here H0{H}_{0}H0​ is the exactly solvable part and H1{H}_{1}H1​ is regarded as the perturbation. To restore the analytical structure of GF, we use the continued fraction to do resummation for the obtained series. The problem of zero-temperature divergence is identified and remedied by the self-consistent series expansion. To demonstrate the implementation of this method, we carry out the weak- as well as the strong-coupling expansion for the Anderson impurity model to order ${\ensuremath{\lambda}}^{2}$. Improved result for the local density of states is obtained by self-consistent second-order strong-coupling expansion and continued fraction resummation.
We explore some further possibilities of application of the projection operator method of Zwanzig to the theory of Green's functions of quantum statistical mechanics, initiated by lchiyanagi, and present a continued fraction representation of the mass operator involving a hierarchy of the random forces. As an application of the theory, we calculate the polarization operator of the phonon Green's function of the Frohlich Hamiltonian in the first approximation which corresponds to the assumption that the electron momenta are orthogonal to the phonon momentum.
Abstract In the previous chapters, we saw how various physical observables can be expressed in terms of retarded Green’s functions. In many cases we need to calculate the time dependence of these functions. There are several ways of attacking this problem, one of which is the equation of motion technique. The basic idea of this method is to generate a series of coupled differential equations by differentiating the function at hand a number of times. If these equations close, the problem is in principle solvable; and if not, one needs to invoke physical arguments to truncate the set of equations in a reasonable fashion. For example, one can neglect certain correlations. We shall study examples of both situations in this chapter.
We present an extension of the analysis previously applied to the retarded part of the coupled cluster (CC) Green's function to its advanced part. In analogy to our earlier studies for the retarded part, we demonstrate that the advanced CC Green's function is expressed in terms of connected diagrams only, which is a direct consequence of algebraic form of equations satisfied by CC amplitudes. We also demonstrate that ω-derivatives of the advanced CC Green's function can be calculated analytically and can be expressed in terms of connected diagrams only. We analyse the structure of connected diagrams and the role of intermediate operators which satisfy electron affinity equation-of-motion CC-type conditions.