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Small distance behavior of one-particle Green's functions in electronic structure theory

Heinz-Juergen Flad, Michael Griebel

TL;DR

The paper develops a rigorous framework for the small-distance analysis of one-particle Green's functions in electronic structure theory by merging singular analysis with pseudo-differential calculus and the limiting absorption principle. It translates spectral resolutions into operator language, constructs parametrices, and uses weighted cone/edge Sobolev spaces to extract leading asymptotics for Green's functions up to second order in perturbation theory, revealing a logarithmic term in certain second-order diagrams. The work shows that partially contracted kernels belong to Hörmander classes like $S^{-3}$ and that their leading singularities follow $H(x_1,x_2,\omega) \sim a_0(x_1,\omega)\ln|x_1-x_2| + a_1(x_1,\omega)|x_1-x_2|+\cdots$, enabling a refined diagrammatic classification by asymptotic smoothness in addition to perturbation order. These insights have potential practical impact by guiding discretization and sparse-grid strategies to reduce computational complexity in many-particle Green's-function calculations.

Abstract

Within the framework of many-particle perturbation theory, we develop an analytical approach that allows us to determine the small distance behavior of Green's functions and related quantities in electronic structure theory. As a case study, we consider the one-particle Green's function up to 2nd order in the perturbation approach. We derive explicit expressions for the leading order terms of the asymptotic small distance behavior. In particular, we demonstrate the appearance of a logarithmic term in the corresponding 2nd order Feynman diagrams. Our asymptotic analysis leads to an improved classification scheme for the diagrams, which takes into account not only the perturbation order, but also the asymptotic smoothness properties near their diagonals. Such a classification may be useful in the design of numerical algorithms and helps to improve their efficiency.

Small distance behavior of one-particle Green's functions in electronic structure theory

TL;DR

The paper develops a rigorous framework for the small-distance analysis of one-particle Green's functions in electronic structure theory by merging singular analysis with pseudo-differential calculus and the limiting absorption principle. It translates spectral resolutions into operator language, constructs parametrices, and uses weighted cone/edge Sobolev spaces to extract leading asymptotics for Green's functions up to second order in perturbation theory, revealing a logarithmic term in certain second-order diagrams. The work shows that partially contracted kernels belong to Hörmander classes like and that their leading singularities follow , enabling a refined diagrammatic classification by asymptotic smoothness in addition to perturbation order. These insights have potential practical impact by guiding discretization and sparse-grid strategies to reduce computational complexity in many-particle Green's-function calculations.

Abstract

Within the framework of many-particle perturbation theory, we develop an analytical approach that allows us to determine the small distance behavior of Green's functions and related quantities in electronic structure theory. As a case study, we consider the one-particle Green's function up to 2nd order in the perturbation approach. We derive explicit expressions for the leading order terms of the asymptotic small distance behavior. In particular, we demonstrate the appearance of a logarithmic term in the corresponding 2nd order Feynman diagrams. Our asymptotic analysis leads to an improved classification scheme for the diagrams, which takes into account not only the perturbation order, but also the asymptotic smoothness properties near their diagonals. Such a classification may be useful in the design of numerical algorithms and helps to improve their efficiency.

Paper Structure

This paper contains 14 sections, 5 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Many-particle models in electronic structure theory
  3. Some basic reduced quantities of many-particle models
  4. Spectral resolutions and their translation back to operator algebras
  5. Singular analysis, the limiting absorption principle and pseudo-differential calculus
  6. Weighted cone and edge Sobolev spaces with asymptotics
  7. The case of conical singularities
  8. The case of edge-type singularities
  9. Asymptotic behavior of one-particle Green's functions
  10. One-particle Green's function in 1st order perturbation theory
  11. 2nd order terms of the one-particle Green's function
  12. Singular behavior of partially contracted kernel functions
  13. Asymptotic smoothness of Feynman diagrams and outlook
  14. Partial contractions of kernel functions

Key Result

Lemma 1

Let $u \in L^{2,s}(\mathbb{R}^3)$ be a function with specific asymptotic behavior, i.e. $u$ belongs to a space ${\cal K}^{s,\gamma}_Q (X^\wedge)$ of asymptotic type $Q$, see Section weightedSobolev. The short-range part of the resolvents $\omega R^{\pm}_0(\lambda)$, $\omega R^{\pm}(\lambda)$, cf. (R with where $\operatorname{\cal P}_{\gamma}, \operatorname{\cal P}_{\gamma+2}, \operatorname{\cal P

Figures (2)

  • Figure 1: (top) The function $\Psi^{(0)}_{0,\beta}$ for $\beta=1000$, and approximations of $\Psi^{(0)}_{0,\infty}$ using Taylor polynomials (\ref{['Taylorvmw']}) of first, second, and third order, respectively. (bottom) The functions $\Psi^{(0)}_{0,\beta}$, for $\beta=1000$, $2000$ and $3000$, compared to $\Psi^{(0)}_{0,\infty}$, obtained from the third order Taylor polynomial (\ref{['Taylorvmw']}).
  • Figure 2: Asymptotic smoothness versus order of perturbation theory for the singular part $F_s$ of certain Feynman diagrams contributing to the one-particle Green's function. The asymptotic smoothness parameter $p$ was given in Definition \ref{['def1']}.

Theorems & Definitions (11)

  • Lemma 1
  • Remark 1
  • Proposition 1
  • Lemma 2
  • proof
  • Example 1
  • Example 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 1 more