The convergence problem of perturbation expansions in the many-body problem is discussed by employing some simple models. The expectation value of rhe resolvent operator is analytic around the origin of the coupling constant so long as the system is enclosed in a finite volume, but as the volume tends to infinity with the density of particles kept constant, the question arises whether this limit is allowed to be taken prior to the sum of the perturbation series. Even if this is the case, the resulting series turns out to be usually divergent, but in some models we consider we can find a function whose asymptotic expansion coincides with this series. It is shown that for the repulsive interactions the function thus obtained gives a correct answer, while for the attractive interactions which lead to a superconducting state it is far from the true one. The asymptotic behavior of the perturbation series in the latter case is examined in some detail.
It is shown analytically and numerically that the finite-temperature many-body perturbation theory in the grand canonical ensemble has zero radius of convergence at zero temperature when the energy ordering or degree of degeneracy for the ground state changes with the perturbation strength. When the degeneracy of the reference state is partially or fully lifted at the first-order Hirschfelder-Certain degenerate perturbation theory, the second-order grand potential and internal energy diverge as $T\ensuremath{\rightarrow}0$. Contrary to earlier suggestions of renormalizability by the chemical potential $\ensuremath{\mu}$, this nonconvergence, first suspected by Kohn and Luttinger, is caused by the nonanalytic nature of the Boltzmann factor ${e}^{\ensuremath{-}E/{k}_{\text{B}}T}$ at
T=0, also plaguing the canonical ensemble, which does not involve $\ensuremath{\mu}$. The finding reveals a fundamental flaw in perturbation theory, which is deeply rooted in the mathematical limitation of power-series expansions and is unlikely to be removed within its framework.
In the conventional perturbation treatment of the many-body problem of interacting particles, the zero-order Hamiltonian corresponds to independent particles moving in a static over-all potential. A discussion of the effects of particle interactions or `correlations' shows that if one starts instead with a Hamiltonian representing noninteracting particles in a velocity-dependent over-all potential, deeper for slow, and shallower for fast particles, then part of the correlation effect is included already in zero order. In addition, a velocity-dependent over-all potential may be called for by velocity dependent interparticle forces or by exchange forces. The degree of the improvement in the convergence of a perturbation expansion based on a Hamiltonian with a velocity-dependent over-all potential is discussed and illustrated by a simple example in which the velocity dependence of the potential gives rise to a reduced "effective mass" of the particles.The many-body problem of a large, uniform system of interacting particles (e.g., the case of a heavy nucleus without surface effects) is formulated in detail in perturbation theory, starting with a velocity-dependent potential constant in space. The work involved in such a calculation turns out to be essentially the same as with a velocity-independent potential, the effect of the velocity dependence being to reduce each term in the perturbation expansion by a constant factor raised to a power equal to the number of "energy denominators" ${E}_{m}\ensuremath{-}{E}_{n}$ in the term in question. A simple equation is deduced for the optimum degree of velocity dependence of the over-all potential which ensures the most rapid convergence.
Realistic effective interparticle interactions of quantum many-body systems are widely seen as being short-range. However, the rigorous mathematical analysis of this type of model turns out to be extremely difficult, in general, with many important fundamental questions remaining open still nowadays. By contrast, mean-field models come from different approximations or Ansätze, and are thus less realistic, in a sense, but are technically advantageous, by allowing explicit computations while capturing surprisingly well many real physical phenomena. Here, we establish a precise mathematical relation between mean-field and short-range models, by using the long-range limit that is known in the literature as the Kac limit. If both attractive and repulsive long-range forces are present then it turns out that the limit mean-field model is not necessarily what one traditionally guesses. One important innovation of our study, in contrast with previous works on the subject, is the fact that we are able to show the convergence of equilibrium states, i.e., of all correlation functions. This paves the way for studying phase transitions, or at least important fingerprints of them like strong correlations at long distances, for models having interactions whose ranges are finite, but very large. It also sheds a new light on mean-field models. Even on the level of pressures, our results go considerably further than previous ones, by allowing, for instance, a continuum of long-range interaction components, as well as very general short-range Hamiltonians for the "free" part of the model.
A perturbation theory is developed for the correlation energy
Ec[n], of a finite-density system, with respect to the coupling constant \ensuremath{\alpha} which multiplies the electron-electron repulsion operator in ${\mathit{H}}^{\mathrm{\ensuremath{\alpha}}}$=T^+\ensuremath{\alpha}V${\mathrm{^}}_{\mathit{e}\mathit{e}}$+
tsumi${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$(
ri). The external potential ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is constrained to keep the gound-state density n fixed for all \ensuremath{\alpha}\ensuremath{\ge}0. ${\mathit{v}}_{\mathrm{\ensuremath{\alpha}}}$ is given completely in terms of functional derivatives at full charge (\ensuremath{\alpha}=1), from which
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]=
ec,2[n]+ ${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}1}$
ec,3[n]+${\ensuremath{\lambda}}^{\mathrm{\ensuremath{-}}2}$
ec,4[n]+..., where each
ec,j[n] is expressed in terms of integrals involving Kohn-Sham determinants. Here, ${\mathit{n}}_{\ensuremath{\lambda}}$(x,y,x)=${\ensuremath{\lambda}}^{3}$n(\ensuremath{\lambda}x,\ensuremath{\lambda}y,\ensuremath{\lambda}z) and \ensuremath{\lambda}=${\mathrm{\ensuremath{\alpha}}}^{\mathrm{\ensuremath{-}}1}$. The identification of ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$], which is a high-density limit, as the second-order energy
ec,2[n] allows one to compute bounds upon ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]; the bounds imply that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$]\ensuremath{\simeq}
Ec[n] for a large class of small atoms and molecules, and suggest that ${\mathrm{lim}}_{\ensuremath{\lambda}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] should be of the same order of magnitude as
Ec[n] in finite insulators and semiconductors.Approximations to
Ec[n] should reflect all this. In contrast, perhaps the well-known overbinding of the local-density approximation (LDA) in molecules and solids is due, in part, to the fact that the LDA correlation energy is too sensitive to a coordinate scaling of n. Indeed, the LDA for
Ec[${\mathit{n}}_{\ensuremath{\lambda}}$] diverges when \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} because of the presence of the -ln(\ensuremath{\lambda}) term in the Gell-Mann and Brueckner high-density expression for the correlation energy, per particle, of a homogeneous density, which is infinite. In a sense, the derived perturbation expansion transforms the Gell-Mann and Brueckner expression into one that applies specifically to an inhomogeneous density which integrates to a finite number of electrons.
A simple upper bound to the ground-state energy of a quantal system composed of
N equal particles obeying Boltzmann or Bose statistics and interacting through the interparticle potential
V(r) is established. This bound applies in the limit of large
N; different expressions obtain depending on whether the potential is or is not singularly attractive at the origin. In the former case, the bound reads - $C{N}^{\frac{(4\ensuremath{-}q)}{(2\ensuremath{-}q)}}$, -
q being the (negative) exponent characterizing the behavior of the pair potential at the origin through ${\mathrm{lim}}_{r\ensuremath{\rightarrow}0}[{r}^{q}V(r)]={V}_{0}$, $\ensuremath{-}\ensuremath{\infty}<{V}_{0}<0$ (of course,
q<2 to prevent two-body collapse); in the latter case, it reads $\ensuremath{-}B{N}^{2}$. Explicit expressions for the constants
C and
B in terms of the pair potential
V(r) are obtained; it is expected (and, in some cases, demonstrated) that the associated upper bound to the ground-state energy of the system approximates closely the actual value. It is also noted that, if for some non-negative value of
p the quantity $\ensuremath{\int}{0}^{\ensuremath{\infty}}\mathrm{dr}{r}^{2} \mathrm{exp}(\ensuremath{-}{p}^{2}{r}^{2})V(r)$ is negative, the
N-body system is unstable in the sense that, at large
N, its total energy is negative and increases in modulus at least as
N2 (so that the binding energy per particle increases in modulus at least as
N). Examples illustrating the nontrivial nature of this conclusion are displayed.