Dyson expansion for form-bounded perturbations, and applications to the polaron problem

TL;DR

This work develops an abstract Dyson expansion for perturbations that are only form-bounded, and applies it to polaron-type Hamiltonians, including the Fröhlich and Nelson models. It yields a rigorous, Wick/Dyson-type expansion for the vacuum heat kernel $\langle \Omega| e^{-t \mathbb{H}(P)} \Omega\rangle$ and a renewal equation, organized by Wick pairings and Dyck paths. Under a complete monotonicity assumption, the map $|P|^2 \mapsto \langle \Omega| e^{-t \mathbb{H}(P)} \Omega\rangle$ is completely monotone, which implies the concavity of the ground-state energy $E_0(P)$ in $|P|^2$ (and, when a ground state exists, strict concavity). The results provide a robust operator-theoretic framework for momentum-dependent spectral properties of a broad class of polaron models, complementing probabilistic approaches and enabling precise structural conclusions such as a renewal equation for the Dyson expansion.

Abstract

We present an abstract Dyson expansion for perturbations that are merely relatively form-bounded, and apply it to the polaron problem. For a large class of polaron-type models, including the Fröhlich and Nelson models, we prove that the vacuum expectation value of the heat semi-group is a completely monotone function of the square of the total momentum. Consequently, the ground state energy is a concave function of the square of the momentum, a result recently proved for the Fröhlich model in \cite{polzer} using a probabilistic approach via Wiener integrals.

Figures (2)

• Figure 1: The contour $\Gamma_{\gamma,\kappa}$.
• Figure 2: Example of a Dyck path of length $2n = 8$, corresponding to $\nu = \{-1,+1,-1,-1,+1,-1,+1,+1\}$.

Theorems & Definitions (7)

• Theorem 1
• Corollary 1
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof