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Wiener-Type Theorems for the Laplace Transform. With Applications to Ground State Problems

Benjamin Hinrichs, Steffen Polzer

TL;DR

The paper develops Wiener-type theorems for the Laplace transform of measures supported near their bottom, expressing atoms at the bottom energy $E$ as ergodic averages of quotients of time-Laplace transforms and relating a second-order term to renewal-theoretic quantities. It connects these analytic formulas to rank-one perturbation theory and renewal theory, yielding a coherent framework to study ground-state existence for finite-dimensional quantum systems coupled to bosonic fields. The authors further apply the framework to generalized Spin-Boson models via a Feynman-Kac representation, obtaining explicit existence and non-existence criteria depending on infrared regularity, decay of correlations, and renewal-transport properties, and providing quantitative bounds on the vacuum overlap. Collectively, the results offer new tools to infer spectral bottom properties and ground-state behavior from Laplace-transform data, with significant implications for quantum field models and their low-energy spectra.

Abstract

We study the behavior of a probability measure near the bottom of its support in terms of time averaged quotients of its Laplace transform. We discuss how our results are connected to both rank-one perturbation theory as well as renewal theory. We further apply our results in order to derive criteria for the existence and non-existence of ground states for a finite dimensional quantum system coupled to a bosonic field.

Wiener-Type Theorems for the Laplace Transform. With Applications to Ground State Problems

TL;DR

The paper develops Wiener-type theorems for the Laplace transform of measures supported near their bottom, expressing atoms at the bottom energy as ergodic averages of quotients of time-Laplace transforms and relating a second-order term to renewal-theoretic quantities. It connects these analytic formulas to rank-one perturbation theory and renewal theory, yielding a coherent framework to study ground-state existence for finite-dimensional quantum systems coupled to bosonic fields. The authors further apply the framework to generalized Spin-Boson models via a Feynman-Kac representation, obtaining explicit existence and non-existence criteria depending on infrared regularity, decay of correlations, and renewal-transport properties, and providing quantitative bounds on the vacuum overlap. Collectively, the results offer new tools to infer spectral bottom properties and ground-state behavior from Laplace-transform data, with significant implications for quantum field models and their low-energy spectra.

Abstract

We study the behavior of a probability measure near the bottom of its support in terms of time averaged quotients of its Laplace transform. We discuss how our results are connected to both rank-one perturbation theory as well as renewal theory. We further apply our results in order to derive criteria for the existence and non-existence of ground states for a finite dimensional quantum system coupled to a bosonic field.

Paper Structure

This paper contains 14 sections, 31 theorems, 230 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Wiener-Type Theorems
  3. Proof of \ref{['Theorem: Convergence of averaged partition function']}
  4. Proof of \ref{['Theorem: Second order term']}
  5. A Link To Rank-One Perturbation Theory
  6. Admissibility of Coupling Constants
  7. Proof of \ref{['Theorem: rank-one perturbation']}
  8. A link to renewal theory
  9. Application to Generalized Spin-Boson Models
  10. Ground State Energy and Existence
  11. Feynman--Kac Formula (\ref{['Proposition: Feynman-Kac']})
  12. Proof of \ref{['Theorem: Estimates Spin-Boson models']}
  13. Proof of \ref{['Theorem: estimate vacuum overlap generalized Spin-Boson', 'Corollary: Existence criterion generalized Spin-Boson']}
  14. Proof of \ref{['Proposition: Estimates in sepcial case of Spin-Boson']}

Key Result

Theorem 2.1

For any $\kappa \in (0, 1)$

Figures (1)

  • Figure 1: Visualization of \ref{['Corollary: classify singularity in E']}. Dormant periods are in blue, active periods are in orange. The more mass the measure has close to $E = \inf \operatorname{supp}(\mu)$, the more the process tends to be dormant. For $\mu = \delta_E$ the process is always dormant, for $\mu(\{E\}) = 0$ and $\int_{(E, \infty)} (x-E)^{-1} \mu(\mathrm dx) < \infty$ the process is eventually active.

Theorems & Definitions (69)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • proof : Proof of \ref{['Theorem: Convergence of averaged partition function']}
  • Proposition 2.5
  • proof
  • proof : Proof of \ref{['Theorem: Second order term']}
  • Remark 3.1
  • ...and 59 more