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On the convergence of the Born series for Coulomb potentials

Ethan Sussman, Jared Wunsch

TL;DR

The paper proves convergence of the Born series for Schrödinger operators with long-range Coulomb-type singularities on non-trapping, asymptotically conic manifolds at high energy, by leveraging Hintz's semiclassical cone estimates and anisotropic vee-Sobolev spaces. It establishes sharp operator-norm bounds for the free resolvent between tailored function spaces, extends these estimates to multiple marked points, and obtains geometric convergence of the Born series with tails that are progressively smoothing. The results enable explicit series formulas for perturbed Coulomb plane waves in short-range, long-range, and dipole regimes, and they extend to the Dyson series for wave propagation, yielding resonance-free regions under suitable hypotheses. Collectively, the work generalizes classical high-energy Born convergence from Euclidean, short-range settings to multi-center Coulomb-type potentials on a broad geometric backdrop, with clear implications for scattering and resonance theory.

Abstract

We provide a short proof of the convergence of the Born series on asymptotically conic manifolds, at sufficiently high energy. The potential is allowed to have multiple Coulomb singularities. This is handled using powerful semiclassical estimates recently proven by Hintz for the case of a single dipole (or better) singularity. The potential is also allowed to be long-range, like the actual Coulomb potential $1/r$; long-range potentials are handled using anisotropic semiclassical (sc-) Sobolev spaces. As a consequence of the above estimates, we show the existence of a resonance-free region for Hamiltonians with multiple screened Coulomb singularities.

On the convergence of the Born series for Coulomb potentials

TL;DR

The paper proves convergence of the Born series for Schrödinger operators with long-range Coulomb-type singularities on non-trapping, asymptotically conic manifolds at high energy, by leveraging Hintz's semiclassical cone estimates and anisotropic vee-Sobolev spaces. It establishes sharp operator-norm bounds for the free resolvent between tailored function spaces, extends these estimates to multiple marked points, and obtains geometric convergence of the Born series with tails that are progressively smoothing. The results enable explicit series formulas for perturbed Coulomb plane waves in short-range, long-range, and dipole regimes, and they extend to the Dyson series for wave propagation, yielding resonance-free regions under suitable hypotheses. Collectively, the work generalizes classical high-energy Born convergence from Euclidean, short-range settings to multi-center Coulomb-type potentials on a broad geometric backdrop, with clear implications for scattering and resonance theory.

Abstract

We provide a short proof of the convergence of the Born series on asymptotically conic manifolds, at sufficiently high energy. The potential is allowed to have multiple Coulomb singularities. This is handled using powerful semiclassical estimates recently proven by Hintz for the case of a single dipole (or better) singularity. The potential is also allowed to be long-range, like the actual Coulomb potential ; long-range potentials are handled using anisotropic semiclassical (sc-) Sobolev spaces. As a consequence of the above estimates, we show the existence of a resonance-free region for Hamiltonians with multiple screened Coulomb singularities.

Paper Structure

This paper contains 14 sections, 5 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of the Born series
  3. Hintz's semiclassical cone estimates
  4. The basic estimate
  5. The estimate for multiple marked points
  6. Convergence of the Born series
  7. Application: series formula for Coulomb plane waves
  8. The short-range case
  9. The long-range case
  10. The dipole case
  11. Application: convergence of the Dyson series
  12. Example of smoothing behavior: $X=\overline{\mathbb{R}^3}$
  13. Various Sobolev spaces
  14. An $L^\infty$ bound on Kummer's confluent hypergeometric function

Key Result

Theorem A

For all $\chi\in C_{\mathrm{c}}^\infty(X^\circ)$ and $\varepsilon,\Sigma>0$, there exists a constant $C>0$ such that the $j$th term $B_j(\sigma \pm i0)$ in the Born series obeys a bound for all $\sigma>\Sigma$. Consequently, if $\sigma$ is large enough, the cutoff Born series $M_\chi\mathsf{B}(\sigma \pm i0)M_\chi$ converges geometrically in the $L^2\to L^2$ operator norm topology to $M_\chi R_V(

Figures (4)

  • Figure 1: The manifold-with-corners $[X\times (0,\infty]_\sigma;\{r=0,\sigma=\infty\}]$ on which the polynomial weights defining the $\vee$-Sobolev spaces are defined. The various boundary hypersurfaces are associated with different Sobolev orders.
  • Figure 3: The sets $S_j=\operatorname{supp}\chi_j$ in the proof of \ref{['thm:main']} are chosen to only contain at most one marked point $p_j$ (blue) each. (They can even be chosen to be disjoint.) In this case, there are $\#=3$ marked points. The set $S_0$ containing $\partial X$ is not shown.
  • Figure 4: The real and imaginary parts of the function $v(s)$, plotted on a log-linear plot vs. $s$. Also shown are the real and imaginary parts of $s^{-i\mathsf{Z}/2\sigma}$, which $v(s)$ approximates at large $s$.
  • Figure 5: A plot of the absolute value of $M(a,1,i\lambda)$ versus $\lambda\in \mathbb{R}$, for $a\in i \mathbb{R}$. \ref{['lem:Kummer_bound']} says that, for each individual $a$, we have an $L^\infty(\mathbb{R}_\lambda)$ bound on $M(a,1,i\lambda)$, with uniformity in $a$ as long as $\operatorname{Im} a$ stays bounded. In §\ref{['sec:multi-Coulomb']}, we care about the $a\to \pm i0$ limit. What we want to emphasize here is that there exist $L^\infty(\mathbb{R}_\lambda)$-bounds uniform in this limit.

Theorems & Definitions (34)

  • Theorem A
  • Remark
  • Remark
  • Remark
  • Example 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Example 3.1
  • proof
  • ...and 24 more