Dyson expansion for form-bounded perturbations, and applications to the polaron problem (2512.13443v1)

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.

• The paper introduces a rigorous Dyson expansion method for unbounded, form-bounded perturbations in self-adjoint quantum semigroups.
• It applies the expansion to polaron models, proving new monotonicity of vacuum expectation values and strict concavity of ground state energies.
• The framework provides robust analytic tools for spectral analysis and paves the way for numerical methods in non-perturbative quantum field theory.

Dyson Expansion for Form-Bounded Perturbations and Applications to the Polaron Problem

Overview

This paper advances the mathematical analysis of quantum dynamics with unbounded perturbations by developing a controlled Dyson expansion for semigroups generated by self-adjoint operators with relatively form-bounded perturbations. The framework is applied to polaron-type Hamiltonians—including both the Fröhlich and Nelson models—revealing new monotonicity and concavity properties for vacuum expectation values and ground state energies as functions of total system momentum. These findings generalize prior momentum-energy concavity results to broad classes of quantum field models and provide new analytic techniques potentially of independent interest for spectral and probabilistic analysis of quantum Bose systems.

Abstract Dyson Expansion for Form-Bounded Perturbations

The Dyson expansion is a standard tool in quantum theory but its rigorous justification typically requires that perturbations be bounded or at least operator-bounded with respect to the unperturbed generator. This paper extends such controlled expansions to the physically relevant case where the perturbation is only form-bounded relative to the unperturbed generator, with relative bound strictly less than one.

Given a non-negative self-adjoint operator $A$ and a hermitian, relatively form-bounded perturbation $B$ (satisfying $$|\langle\psi|B\psi\rangle|\leq \lambda(\langle\psi|A\psi\rangle + a\|\psi\|^2)$$ for some $0<\lambda<1$), the expansion

$e^{-t(A+B)} = e^{-tA} + \sum_{n\geq 1} D_n,$

with explicit analytic expressions for the $D_n$, is established. The terms are shown to be well-defined and the expansion convergent in operator norm. This is achieved through a non-trivial contour integral approach relying on the resolvent identity and careful spectral estimates, circumventing the need for the stronger operator-boundedness condition. The resulting bounds on the expansion's terms decay rapidly with $n$, guaranteeing uniform convergence for all $t > 0$. This technical innovation is a foundational tool for all subsequent results.

Applications to the Polaron Model

The central physical application is to the momentum-resolved polaron Hamiltonian of the general form:

$$\mathbb{H}(P) = |P-\mathbb{P}_f|^2 + d\Gamma(\omega) + \Phi(v),$$

acting in a symmetric Fock space, with $P$ the conserved total momentum, $\mathbb{P}_f$ the field momentum, $d\Gamma(\omega)$ the free field Hamiltonian, and $\Phi(v)$ a field operator arising from particle-boson interactions. Under mild integrability and support conditions on the coupling $v$ and the field dispersion $\omega$, verified using the commutator method of Lieb–Yamazaki, the paper proves that $\Phi(v)$ is infinitesimally form-bounded with respect to the free Hamiltonian and thus the entire Hamiltonian admits the functional analytic control needed for the abstract Dyson expansion.

Dyson Series for Vacuum Expectations

The core mathematical object is the vacuum expectation value of the semigroup, $\langle\Omega| e^{-t\mathbb{H}(P)}|\Omega\rangle$, with $\Omega$ the Fock vacuum. The authors provide an explicit, convergent Dyson series for this object. The combinatorics of the expansion are codified via sets of Wick pairings and Dyck paths, mapping directly to pair creation and annihilation processes at the field level. Each term in the series is expressed as an explicit, finite-dimensional, positive oscillatory integral over time and momentum variables. The expansion makes rigorous an expression that, for unbounded perturbations, has not previously been available beyond the formal level.

Renewal Equation and its Consequences

Under additional assumptions of rotational invariance and monotonicity (inspired by properties of Laplace transforms of positive measures and applied in both Fröhlich and Nelson models), the vacuum expectation is shown to satisfy a renewal equation of the form

$$\langle\Omega | e^{-t\mathbb{H}(P)} |\Omega\rangle = e^{-t|P|^2} + \int_0^t \langle\Omega | e^{-(t-s)\mathbb{H}(P)} |\Omega\rangle\, f_P(s) ds,$$

where $f_P(s)$ is given by an explicit sum/integral involving so-called interlacing Wick pairings. This equation structurally encodes all repeated interactions with the field in a memory kernel formalism, facilitating further analytic properties through renewal theory.

Monotonicity and Concavity Results

A key result is that $$|P|^2 \mapsto \langle\Omega|e^{-t\mathbb{H}(P)}|\Omega\rangle$$ is a completely monotonic function for all $t > 0$. By Bernstein's theorem, this means it is the Laplace transform of a positive measure, imparting strong regularity and sign-definite properties to the function. Such monotonicity is new in its generality for polaron-type systems.

The main physical implication is for the ground state energy $E_0(P) = \inf\operatorname{spec} \mathbb{H}(P)$, which is shown to be a concave function of $|P|^2$ (and even strictly concave where ground states exist). This sharply delimits the possible dispersion relations for polarons and excludes scenarios where momentum increases could lead to anomalous increases in ground state energies. The result significantly strengthens previously known facts, some of which were available only via path integral probabilistic techniques for special cases [see Polzer, Lett. Math. Phys. 113, 90 (2023)], by providing operator-theoretic and analytic proofs directly from the semigroup framework.

Theoretical and Practical Implications

The theoretical advances facilitate rigorous spectral analysis of polaron models and similar quantum systems with merely form-bounded interactions. The strict concavity of the ground state energy excludes unexpected energy-momentum behavior and supports the physical understanding of polaron effective mass increasing with coupling, as typically anticipated.

On the technical side, the abstract Dyson expansion for form-bounded perturbations opens prospects for perturbative and non-perturbative investigations in broader QFT, potentially impacting studies of infrared problems, effective mass renormalization, and spectral gap questions. The renewal equation structure may also inspire new probabilistic representations and path integral methods for interacting fields beyond the Gaussian setup.

From a practical perspective, the analytic expansion and associated estimates can in principle be used for controlled numerical approaches for correlation functions and spectral quantities, particularly in non-perturbative regimes where path integral representations are intractable.

Conclusion

This paper rigorously establishes a Dyson expansion for dynamics generated by self-adjoint operators perturbed by merely form-bounded terms and applies it to polaronic Hamiltonians, demonstrating complete monotonicity of vacuum expectations and strict concavity of the energy-momentum relation. These results generalize and strengthen earlier findings in mathematical physics, enable new approaches to non-perturbative quantum field analysis, and clarify the structure of important models in particle-boson interaction theory.

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Reference: "Dyson expansion for form-bounded perturbations, and applications to the polaron problem" (2512.13443)