Convergent perturbation theory for Coulomb scattering and charged few-body systems: the uncompleted core of Weinberg's 1963 quasiparticle program

Weinberg's 1963 paper 'Quasiparticles and the Born Series' proved that the Born (Neumann) series for two-body nonrelativistic scattering converges at a complex energy $W$ off the positive real axis if and only if every characteristic eigenvalue $\eta_\nu(W)$ of the Lippmann–Schwinger kernel $[W-H_0]^{-1}V$ lies inside the unit circle, and showed that divergences caused by eigenvalues outside it — bound states, resonances, and composite particles — can be removed by separable 'quasiparticle' subtractions, restoring an always-convergent perturbation expansion for short-range interactions. The paper explicitly left open the extension beyond two-body processes, the constructive determination of the required vertices, the zero-energy threshold $W=0$ and long-range (Coulomb) forces, and a decision procedure for convergence on the positive-energy scattering axis. The short-range parts of that program were subsequently closed — by Weinberg's own 1964 multiparticle sequel, the connected-kernel (Faddeev-type) few-body theory, the separable-expansion program, and a 2026 guaranteed-convergent 'Weinberg-regularized' algorithm (see Background for the supporting works) — while the long-range core remains open: there is no rigorous, constructive convergence theory for Coulomb-type interactions at arbitrary fixed positive energies (including the $W\to 0$ threshold), where on-shell Born terms of second and higher order diverge, nor for scattering of three or more charged particles. The open problem is to complete the quasiparticle program in precisely those regimes.

Background

Nonrelativistic two-body scattering is governed by the Lippmann–Schwinger equation $T(W)=V+V[W-H_0]^{-1}T(W)$ for the transition (scattering) operator $T$, where $H_0$ is the free Hamiltonian, $V$ the interaction, and $W$ a complex energy. Iterating yields the Born series (the Neumann iteration of the integral equation, $T=V+VG_0V+VG_0VG_0V+\cdots$ with $G_0=[W-H_0]^{-1}$), the scattering-theoretic analogue of perturbation theory; physical scattering amplitudes are the on-shell matrix elements of $T(W)$ at $W=E+i0$ on the positive-energy (scattering) axis. For weak interactions the iteration converges, but for strong interactions — in particular near bound states and resonances — it generally diverges, and deciding convergence at a given energy and coupling is a classical problem.

For interactions of short range (potentials decaying faster than $1/r$), the kernel $[W-H_0]^{-1}V$ is a compact operator on the relevant Hilbert space — intuitively, it is sufficiently smoothing and decaying that Fredholm determinant theory for integral equations applies, and its spectrum is a discrete set of characteristic eigenvalues $\eta_\nu(W)$ accumulating only at zero (now usually called Weinberg eigenvalues). Weinberg's 1963 paper 'Quasiparticles and the Born Series' proved that, off the positive real axis, the Born series converges if and only if every $\eta_\nu(W)$ lies inside the unit circle; the radius of convergence in a coupling constant $\lambda$ scaling $V\to\lambda V$ is $1/|\eta_1(W)|$, where $\eta_1$ is the eigenvalue of largest modulus. Eigenvalues outside the unit circle correspond to bound states, resonances, or states that would be bound for a rescaled interaction — collectively, composite particles. The central construction removes the offending spectral parts by separable subtractions, $V_1(W)=V-V|\Gamma(W)\rangle\langle\bar{\Gamma}(W)|V$, where the vertices $|\Gamma\rangle$ approximate the eigenvectors belonging to eigenvalues outside the unit circle; the subtracted pieces are resummed exactly (summed in closed form) and interpreted as fictitious elementary excitations ('quasiparticles' — a device of scattering theory, not the Landau quasiparticle of many-body physics). Under $L^2$/trace conditions (square-integrable kernels with finite traces) only finitely many subtractions are needed at any given energy — with an explicit exception at zero energy for long-range forces — so the reduced Born series always converges. Byproducts included operator-level proofs of the applicability of the Fredholm theorems, of the Pais–Jost identity between the Fredholm determinant and the Jost function (the entire function whose zeros encode the bound states), of Levinson's theorem (which fixes the zero-energy scattering phase shift in terms of the number of bound states), of the Bargmann–Schwinger bound on the number of bound states, of convergence of the Born series at high energy, and first-order quasiparticle formulas that, for example, reproduce the exact hydrogen ground-state binding energy.

The paper was explicit that this did not complete the program, acknowledging four limitations: (i) the trace/Fredholm methods fail for three-body and higher processes, where the analogous traces become infinite; (ii) no general algorithmic prescription yields the required vertices and Fredholm coefficients in the most general cases (those not reducible to a potential kernel or to a Volterra — triangular — kernel that iterates term by term in closed form); (iii) the zero-energy case $W=0$ and long-range forces, notably the Coulomb potential $V(r)\propto 1/r$, require special handling, and a fully rigorous treatment of the limit to the physical cut for long-range interactions was not completed; (iv) beyond sufficient trace/sum-rule criteria at negative energies, no simple, generally applicable decision procedure determines convergence of the original Born series on the positive-energy scattering axis for arbitrary interactions.

The subsequent history closes the short-range parts of this list. The multiparticle obstruction was precisely identified and overcome by Weinberg himself in 1964: the multiparticle Lippmann–Schwinger kernel is not $L^2$ — it has a continuous spectrum, producing cuts in the coupling-constant plane for multiparticle amplitudes — but equations built from connected graphs only do have $L^2$ kernels and are solvable by Fredholm/quasiparticle methods; the systematic N-particle reduction was consolidated by Alt, Grassberger, and Sandhas (1967) and reviewed by Joachain and Quigg (1974). Constructive separable expansions of the two-body transition matrix became a standard industry, and in 2026 a 'Weinberg-regularized' iterative algorithm with guaranteed convergence was demonstrated for short-range multichannel molecular scattering. Weinberg eigenvalues computed at positive energies are now the standard numerical diagnostic of perturbativeness in nuclear physics. What was never completed is the long-range core: for Coulomb-type interactions the kernel is not compact, on-shell Born terms of second and higher order diverge individually, the threshold $W\to 0$ is singular, and for three or more charged particles no compactness theorem for the connected equations is established — current few-body practice screens the Coulomb interaction (replacing it by a truncated short-range potential) and renormalizes (undoes the screening) without a general convergence proof, and the first rigorous convergence result for long-range tails and Coulomb singularities (2025) holds only in the high-energy limit.

Problem Statement

Complete the convergence theory of the Born series and its quasiparticle (separable) regularization for nonrelativistic scattering with long-range, Coulomb-type interactions — the regimes explicitly excluded in 1963 that later work has still not closed. Concretely, for interactions with Coulomb-type behavior (tails decaying like $1/r$ at large distances, possibly with several Coulomb singularities):

1. Two-body, all positive energies. Establish a rigorous convergence theory of the Born series at fixed physical energies $W=E+i0$, for interactions of arbitrary coupling strength — beyond the regimes already settled (sufficiently weak coupling, and the high-energy limit proved by Sussman and Wunsch (2025)) — including the threshold limit $W\to 0$. This comprises (a) a criterion that decides, for the stated potential class and fixed positive energy (including the threshold limit $W\to 0$), whether the Born series converges for the on-shell amplitudes under an admissible regularization that preserves the Coulomb $1/r$ tail rather than replacing the interaction by a short-range, screened, or otherwise surrogate kernel, playing the role that the eigenvalue condition $|\eta_\nu(W)|<1$ plays off the axis; and (b) a constructive quasiparticle-type separable regularization whose convergence is proven at every such energy, with controlled, computable approximation error — extending to long-range forces what the 2026 Weinberg-regularized algorithms guarantee only for short-range interactions.

2. Charged multiparticle systems. Extend the connected-kernel Fredholm/quasiparticle theory — the mechanism by which Weinberg (1964) and the Alt-Grassberger-Sandhas reduction turned short-range three- and more-particle scattering into solvable $L^2$-kernel equations — to systems of three or more particles with Coulomb interactions: prove compactness, or a replacement property stated explicitly that yields the same Fredholm/quasiparticle consequences for the connected equations, for the full Coulomb interaction rather than a screened or truncated surrogate, at all energies (going beyond the screening-and-renormalization practice and partial compactness claims in the literature), and construct from it a convergent expansion for the scattering amplitudes with a computable error bound.

These two deliverables are facets of one objective — establishing when and how perturbation theory can be made convergent for Coulombic scattering, completing Weinberg's claim that perturbation theory 'always works' once composite particles are replaced by quasiparticles — and they share the same obstruction: the Lippmann-Schwinger kernel is not compact for $1/r$ tails, and its spectrum is singular on the positive-energy axis and at $W=0$. Partial progress on either facet (for example, a fixed-energy two-body criterion, or a compactness theorem for a named class of three-charge configurations) is meaningful on its own, but the program is complete only when both are settled in the generality stated.

References

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