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Fumio Hiroshima

Publications and source records attributed to Fumio Hiroshima.

At least 19 recordsLinked to original sources
arXiv2026

Lower bounds on the spatial decay of the ground state of the Pauli-Fierz model

Lower bounds on the pointwise spatial decay of the ground state of the Pauli-Fierz model in non-relativistic quantum electrodynamics are studied. Two models are considered: the full Pauli-Fierz model and the Pauli-Fierz model under the dipole approximation. For the full model, a lower bound is derived by means of a probabilistic approach based on the Feynman-Kac formula. The application of the Agmon distance method to the full model encounters a serious difficulty arising from the path dependence of a double stochastic integral. In contrast, under the dipole approximation, the corresponding integrand becomes deterministic, which allows a lower bound to be obtained in terms of the Agmon distance.

math-ph↗
arXiv2026

Strong-coupling asymptotics for anisotropic Rabi model

The anisotropic quantum Rabi model provides a continuous interpolation between the quantum Rabi model and the Jaynes-Cummings model. In this paper, we study the norm-resolvent limit of the anisotropic quantum Rabi model as the coupling strength tends to infinity. After subtracting an explicit quadratic renormalization term, we identify the limiting Hamiltonian, which determines the asymptotic spectral structure of the model. Our result provides a unified description connecting the Jaynes-Cummings model and the full quantum Rabi model in the strong-coupling limit, thereby revealing the dominant role of the counter-rotating interaction.

math-ph↗
arXiv2026

The weak coupling limit of the Pauli-Fierz model

We investigate the weak coupling limit of the Pauli- Fierz Hamiltonian within a mathematically rigorous framework. Furthermore, we establish the asymptotic behavior of the effective mass in this regime.

math-ph↗
arXiv2025

Representations of Josephson junction on the unit circle and the derivations of Mathieu operators and Fraunhofer patterns

The Hamiltonian J of the Josephson junction is introduced as a self-adjoint operator on l2 tensor l2. It is shown that J can also be realized as a self-adjoint operator HS1 on L2(S1) tensor L2(S1), from which a Mathieu operator given by "-d^2/dθ^2 - 2α cos θ" is derived. A fiber decomposition of HS1 with respect to the total particle number is established, and the action on each fiber is analyzed. In the presence of a magnetic field, a phase shift defines the magnetic Josephson junction Hamiltonian HS1(Φ) and the Josephson current IS1(Φ). For a constant magnetic field inducing a local phase shift Φ(x), the corresponding local current IS1(Φ(x)) is computed, and it is proved that the Fraunhofer pattern arises naturally.

math-ph↗
arXiv2025

Conjugate Operators of 1D-harmonic Oscillator

A conjugate operator $T$ of one-dimensional harmonic oscillator $N$ is defined by an operator satisfying canonical commutation relation $[N,T]=-i\one$ on some domain but not necessarily a dense one. Examples of conjugate operators include the angle operator $\TA$ and the Galapon operator $\TG$. Let $\sT$ denote a set of conjugate operators of $N$ of the form $T_{ω,m}=\frac{i}{m}\log(ω\one-L^m)$ with $(ω, m)\in \overline{\DD}\times (\NN\setminus\{0\})$, where $L$ is a shift operator and $\DD$ denotes the open unit disc in the complex plane $\CC$. A classification of $\sT$ is given as $\sT=\sT_{\{0\}}\cup\sT_{\DD\setminus\{0\}}\cup \sT_{\partial \DD}$, where $\TA\in\sT_{\{0\}}$ and $\TG\in \sT_{\partial \DD}$. The classification is specified by a pair of parameters $(\om,m)\in\CC\times\NN$. Finally the time evolution $T_{\om,m}(t)=e^{itN} T_{\om,m}e^{-itN}$ for $T_{\om,m}\in\sT$ is investigated, and it is shown that $T_{\om,m}(t)$ is periodic with respect to~$t$.

math-ph↗
arXiv2025

Ordering of Energy Levels in the Fröhlich Model

Consider a one-dimensional system of \( N \) electrons subject to an external potential \( U \). Let \( E_{\rm el}(S) \) denote the ground state energy of the system with total spin \( S \). The Mattis--Lieb theorem asserts that, for a broad class of potentials \( U \), the inequality \( E_{\rm el}(S) < E_{\rm el}(S') \) holds whenever \( S < S' \). This result implies that the ground state of a one-dimensional many-electron system is non-ferromagnetic. In the present work, we demonstrate that the Mattis--Lieb theorem can be extended to electron-phonon interacting systems governed by the Fröhlich model. Our analysis is carried out in the setting without an ultraviolet cutoff. The cornerstone of our approach is the construction of a Feynman--Kac-type formula for the heat semigroup generated by the Fröhlich Hamiltonian.

math-ph↗
arXiv2025

Fiber decomposition of non-commutative harmonic oscillators by two-photon quantum Rabi models

The non-commutative harmonic oscillators (NcHO) and 2p-quantum Rabi models (2pQRM) are extensions of harmonic oscillators. The purpose of this paper is to give a relationship between NcHO and 2pQRM, and the fiber decomposition of NcHO by 2pQRM is shown. We also construct Feynman-Kac formulas of NcHO and 2pQRM. Then asymptotic behaviors of the spectral zeta function of 2pQRM is considered.

math-ph↗
arXiv2025

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

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.

math-ph↗
arXiv2024

Spectral zeta function and ground state of quantum Rabi model

The spectral zeta function of the quantum Rabi Hamiltonian is considered. It is shown that the spectral zeta function converges to the Riemann zeta function as the coupling constant goes to infinity. Moreover the path measure associated with the ground state of the quantum Rabi Hamiltonian is constructed on a discontinuous path space, and several applications are shown.

math-ph↗
arXiv2024

On the Ergodicity of Renormalized Translation-Invariant Nelson-Type Semigroups

We present a simple functional integration based proof that the semigroups generated by the ultraviolet-renormalized translation-invariant non- and semi-relativistic Nelson Hamiltonians are positivity improving (and hence ergodic) with respect to the Fröhlich cone for arbitrary values of the total momentum. Our argument simplifies known proofs for ergodicity and the result is new in the semi-relativistic case.

math-ph↗
arXiv2024

Self-adjointness of unbounded time operators

Time operators for an abstract semi-bounded self-adjoint operator $H$ with purely discrete spectrum is considered. The existence of a bounded self-adjoint time operator $T$ for $H$ is known as Galapon time operator. In this paper, a self-adjoint but unbounded time operator $T$ of $H$ is constructed.

math-ph↗
arXiv2024

Time Operators of Harmonic Oscillators and Their Representations

A time operator $\hat T_\eps$ of the one-dimensional harmonic oscillator $ \hat h_\eps=\half(p^2+\eps q^2)$ is rigorously constructed. It is formally expressed as $ \hat T_\eps=\half\frac{1}{\sqrt \eps } (\arctan (\sqrt \eps \hat t_0)+\arctan (\sqrt \eps \hat t_1))$ with $\hat t_0=p^{-1}q$ and $\hat t_1=qp^{-1}$. It is shown that the canonical commutation relation $[h_\eps, \hat T_\eps ]=-i\one$ holds true on a dense domain in the sense of sesqui-linear forms, and the limit of $\hat T_\eps $ as $\eps\to 0$ is shown. Finally a matrix representation of $\hat T_\eps$ and its analytic continuation are given.

math-ph↗
arXiv2022

Towards a derivation of Classical ElectroDynamics of charges and fields from QED

The purpose of this article is twofold. On one hand, we rigorously derive the Newton--Maxwell equation in the Coulomb gauge from first principles of quantum electrodynamics in agreement with the formal Bohr's correspondence principle of quantum mechanics. On the other hand, we establish the global well-posedness of the Newton--Maxwell system on energy-spaces under weak assumptions on the charge distribution. Both results improve the state of the art, and are obtained by incorporating semiclassical and measure theoretical techniques. One of the novelties is the use of quantum propagation properties in order to build global solutions of the Newton--Maxwell equation.

math.AP↗
arXiv2021

Ground states and associated path measures in the renormalized Nelson model

We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson's operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

math-ph↗
arXiv2020

Functional central limit theorems and $P(phi)_{1}$-processes for the classical and relativistic Nelson models

We construct $P(phi)_1$-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for additive functionals of these processes. We discuss a number of examples by choosing specific functionals related to particle-field operators.

math-ph↗
arXiv2019

Spectrum of the semi-relativistic Pauli-Fierz model II

We consider the semi-relativistic Pauli-Fierz Hamiltonian $$ H_m = |{\bf p}-{\bf A}({\bf x})| + H_{f,m} + V({\bf x}),\quad m\geq0, $$ and prove the existence of the ground state of $H_m$ for $m=0$. Here ${\bf A}({\bf x})$ denotes a quantized radiation field and $H_{f,m}$ the free field Hamiltonian with the dispersion relation $\sqrt{|{\bf k}|^2+m^2}$ with $m\geq0$. This paper is the sequel of [HH16], where the existence of the ground state $Φ_m$ of $H_m$ for $m>0$ is proven. In order to show the existence of the ground state for $m=0$ we estimate a singular and non-local pull-through formula and show the equicontinuity of set $\{a(k)Φ_m\}_{0<m<m_0}$ with some $m_0$, where $a(k)$ denotes the formal kernel of the annihilation operator. Taking a subsequence $m_j$, we can conclude that $\lim_{m_j\to0}Φ_{m_j}=Φ_0\not=0$ and $Φ_0$ is the ground state of $H_0$.

math-ph↗
arXiv2017

Mass Renormlization in the Nelson Model

The asymptotic behavior of the effective mass $m_{\rm eff}(Λ)$ of the so-called Nelson model in quantum field theory is considered, where $Λ$ is an ultraviolet cutoff parameter of the model. Let $m$ be the bare mass of the model. It is shown that for sufficiently small coupling constant $|α|$ of the model, $m_{\rm eff}(Λ)/m$ can be expanded as $m_{\rm eff}(Λ)/m= 1+\sum_{n=1}^\infty a_n(Λ) α^{2n}$. A physical folklore is that $a_n(Λ)\sim [\log Λ]^{(n-1)}$ as $Λ\to \infty$. It is rigorously shown that $$0<\lim_{Λ\to\infty}a_1(Λ)<C,\quad C_1\leq \lim_{Λ\to\infty}a_2(Λ)/\logΛ\leq C_2$$ with some constants $C$, $C_1$ and $C_2$.

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