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Two-sided bounds on the point-wise spatial decay of ground states in the renormalized Nelson model with confining potentials

Fumio Hiroshima, Oliver Matte

TL;DR

The paper derives two-sided, point-wise bounds on ground-state decay for the renormalized Nelson model with confining potentials by marrying Feynman–Kac representations with Agmon-distance analysis. The authors establish a lower bound that matches previously known upper bounds in leading order for a broad class of confining potentials, demonstrating that the ground-state decay is governed by the same Agmon metric as in Schrödinger operators, even in a quantum field theoretic setting without ultraviolet regularization. The approach relies on a detailed FK framework, exponential moment estimates, and a careful treatment of path space variational problems, including positivity results for the ground state. This work advances understanding of spatial localization in quantum field theories and highlights the universality of Agmon-type decay rates across particle-field models, with potential implications for spectral analysis and stability in non-relativistic QFT models.

Abstract

We study the renormalized Nelson model for a scalar matter particle in a continuous confining potential interacting with a possibly massless quantized radiation field. When the radiation field is massless we impose a mild infrared regularization ensuring that the Nelson Hamiltonian has a non-degenerate ground state in all considered cases. Employing Feynman-Kac representations, we derive lower bounds on the point-wise spatial decay of the partial Fock space norms of ground state eigenvectors. Here the exponential rate function governing the decay is given by the Agmon distance familiar from the analysis of Schrödinger operators. For a large class of confining potentials, our lower bounds on the decay of ground state eigenvectors match asymptotically with the upper bounds implied by previous work of the present authors.

Two-sided bounds on the point-wise spatial decay of ground states in the renormalized Nelson model with confining potentials

TL;DR

The paper derives two-sided, point-wise bounds on ground-state decay for the renormalized Nelson model with confining potentials by marrying Feynman–Kac representations with Agmon-distance analysis. The authors establish a lower bound that matches previously known upper bounds in leading order for a broad class of confining potentials, demonstrating that the ground-state decay is governed by the same Agmon metric as in Schrödinger operators, even in a quantum field theoretic setting without ultraviolet regularization. The approach relies on a detailed FK framework, exponential moment estimates, and a careful treatment of path space variational problems, including positivity results for the ground state. This work advances understanding of spatial localization in quantum field theories and highlights the universality of Agmon-type decay rates across particle-field models, with potential implications for spectral analysis and stability in non-relativistic QFT models.

Abstract

We study the renormalized Nelson model for a scalar matter particle in a continuous confining potential interacting with a possibly massless quantized radiation field. When the radiation field is massless we impose a mild infrared regularization ensuring that the Nelson Hamiltonian has a non-degenerate ground state in all considered cases. Employing Feynman-Kac representations, we derive lower bounds on the point-wise spatial decay of the partial Fock space norms of ground state eigenvectors. Here the exponential rate function governing the decay is given by the Agmon distance familiar from the analysis of Schrödinger operators. For a large class of confining potentials, our lower bounds on the decay of ground state eigenvectors match asymptotically with the upper bounds implied by previous work of the present authors.
Paper Structure (14 sections, 11 theorems, 82 equations)

This paper contains 14 sections, 11 theorems, 82 equations.

Table of Contents

  1. Introduction and main result
  2. General introduction
  3. The Nelson Hamiltonian
  4. Spatial decay of ground state eigenvectors
  5. Feynman--Kac formula and some of its implications
  6. Some bosonic Fock space calculus
  7. Feynman--Kac representation
  8. Boundedness and continuity in the range of the semigroup
  9. Proof of the upper bounds on the spatial decay
  10. Positivity of ground state eigenvectors
  11. Point-wise lower bound on the spatial decay
  12. A general lower bound on the spatial decay
  13. Optimal paths and geodesic distance
  14. Basic properties of the geodesic distance

Key Result

Proposition 1.1

Assume that $V(x)\to\infty$ as $|x|\to\infty$. Further, assume the following infrared regularity condition to hold: Then $E_{\rm g}$ is a non-degenerate eigenvalue of $H$.

Theorems & Definitions (27)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Theorem 1.3
  • proof
  • Remark 1.4
  • Example 1.5
  • Example 1.6
  • Remark 1.7
  • Remark 1.8
  • ...and 17 more