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Asymptotic series for low-energy excitations of the Fröhlich Polaron at strong coupling

Morris Brooks, David Mitrouskas

TL;DR

This work analyzes the Fröhlich polaron in a confined region under strong coupling, proving that each low-energy eigenvalue admits an asymptotic expansion in inverse powers of the coupling constant $\alpha$ with leading Pekar energy $e^{Pek}$. The authors develop a two-stage perturbative framework: first around the Pekar minimizer to obtain a Bogoliubov-type description of quantum field fluctuations, then around its low-energy excitations to extract higher-order energy corrections $(E_0,E_1,\dots)$. They provide explicit recursive formulas for the coefficients via matrices $M^{(\ell)}$ built from the interaction structure and reduced resolvents, and treat degeneracies with a general matrix perturbation approach. Additionally, they derive improved remainder estimates through optimized approximate eigenstates and discuss connections to potential Borel summability, highlighting a rigorous adiabatic-like decoupling between electron and field in the strong-coupling regime.

Abstract

We consider the confined Fröhlich polaron and establish an asymptotic series for the low-energy eigenvalues in negative powers of the coupling constant. The coefficients of the series are derived through a two-fold perturbation approach, involving expansions around the electron Pekar minimizer and the excitations of the quantum field.

Asymptotic series for low-energy excitations of the Fröhlich Polaron at strong coupling

TL;DR

This work analyzes the Fröhlich polaron in a confined region under strong coupling, proving that each low-energy eigenvalue admits an asymptotic expansion in inverse powers of the coupling constant with leading Pekar energy . The authors develop a two-stage perturbative framework: first around the Pekar minimizer to obtain a Bogoliubov-type description of quantum field fluctuations, then around its low-energy excitations to extract higher-order energy corrections . They provide explicit recursive formulas for the coefficients via matrices built from the interaction structure and reduced resolvents, and treat degeneracies with a general matrix perturbation approach. Additionally, they derive improved remainder estimates through optimized approximate eigenstates and discuss connections to potential Borel summability, highlighting a rigorous adiabatic-like decoupling between electron and field in the strong-coupling regime.

Abstract

We consider the confined Fröhlich polaron and establish an asymptotic series for the low-energy eigenvalues in negative powers of the coupling constant. The coefficients of the series are derived through a two-fold perturbation approach, involving expansions around the electron Pekar minimizer and the excitations of the quantum field.
Paper Structure (14 sections, 26 theorems, 191 equations)

This paper contains 14 sections, 26 theorems, 191 equations.

Table of Contents

  1. Introduction and main results
  2. Outline of the proof
  3. Spectrum of the Bogoliubov Hamiltonian
  4. Two-term expansion of low-energy eigenvalues
  5. Preliminary estimates
  6. Upper bounds
  7. Lower bounds
  8. Asymptotic series for low-energy eigenvalues
  9. Gross transformed Hamiltonian
  10. Approximate eigenstates
  11. Proofs of Theorems \ref{['thm:asymptotic:expansion:1']} and \ref{['thm:recursive:formula:generalcase']}
  12. Improved remainder estimates
  13. Proof of Proposition \ref{['prop:asym:expansion:HG']}
  14. Proof of Lemma \ref{['lem:estimates:approx:groundstate:2']}

Key Result

Theorem 1.2

For $n\in \mathbb N$ let $\mathscr E^{(n)}(\alpha)$ denote the $n th$ eigenvalue of the Fröhlich Hamiltonian $\mathfrak H_\alpha$ and $\mathsf E^{(n)}$ the $n th$ eigenvalue of the Bogoliubov Hamiltonian $\mathbb H_0$. There exists a sequence $(E_\ell )_{\ell \in \mathbb N_0 }$ with $E_0 = \mathsf E for all $\alpha \ge \alpha(b)$.

Theorems & Definitions (52)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 1.9
  • proof
  • ...and 42 more