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Convergence of states for polaron models in the classical limit

Marco Falconi, Alessandro Olgiati, Nicolas Rougerie

Abstract

We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model's energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.

Convergence of states for polaron models in the classical limit

Abstract

We consider the quasi-classical limit of Nelson-type regularized polaron models describing a particle interacting with a quantized bosonic field. We break translation-invariance by adding an attractive external potential decaying at infinity, acting on the particle. In the strong coupling limit where the field behaves classically we prove that the model's energy quasi-minimizers strongly converge to ground states of the limiting Pekar-like non-linear model. This holds for arbitrarily small external attractive potentials, hence this binding is fully due to the interaction with the bosonic field. We use a new approach to the construction of quasi-classical measures to revisit energy convergence, and a localization method in a concentration-compactness type argument to obtain convergence of states.
Paper Structure (17 sections, 15 theorems, 210 equations)

This paper contains 17 sections, 15 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Model
  4. Statements
  5. Organization of the paper
  6. Reduced densities, localization of states and localization of energies
  7. Reduced densities
  8. Localization of states
  9. Energy localization
  10. Quasi-classical measures
  11. Notation
  12. The theorem
  13. Proof of Theorem \ref{['thm:deF comp']}
  14. Convergence of the energy
  15. Generalized Pekar energies
  16. ...and 2 more sections

Key Result

Theorem 2.3

Let $\Psi_\alpha \in\mathfrak{H}$ be a (family of) normalized vector(s) such that Let $k,\ell$ be non-negative integers with $k+\ell \leq 2$. Modulo extraction of a subsequence in $\alpha$, for every bounded $A\in \mathcal{B}( L^2_\mathrm{part}(\mathbb{R}^d) )$ and for every $f_1,\dots,f_k,g_1,\dots,g_\ell\in L^2_\mathrm{field}(\mathbb{R}^d)$, where $P$ is a probability measure over the set of P

Theorems & Definitions (30)

  • Theorem 2.3: Convergence of states in the quasi-classical limit
  • Definition 3.1: Reduced density matrices for particle and field
  • Lemma 3.2: Field-particle reduced density matrix
  • proof
  • Proposition 3.3: Construction of localized states
  • proof
  • Proposition 3.4: Energy localization
  • Lemma 3.5: Particle energy localization
  • proof
  • Lemma 3.6: Field energy localization
  • ...and 20 more