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Fumio Hiroshima

Publications and source records attributed to Fumio Hiroshima.

At least 37 records · Page 2Linked to original sources
arXiv2017

Ultra-Weak Time Operators of Schroedinger Operators

In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schroedinger operators H with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions $f$ such that $f(H)$ has an ultra-weak time operator is found.

math-ph↗
arXiv2017

Kato's Inequality for Magnetic Relativistic Schrödinger Operators

Kato's inequality is shown for the magnetic relativistic Schrödinger operator $H_{A,m}$ defined as the operator theoretical {\it square root} of the selfadjoint, magnetic nonrelativistic Schrödinger operator $(-i\nabla-A(x))^2+m^2$ with an $L^{2}_{\text{\rm loc}}$ vector potential $A(x)$.

math.AP↗
arXiv2015

Periodic Solutions of Generalized Schrödinger Equations on Cayley Trees

In this paper we define a discrete generalized Laplacian with arbitrary real power on a Cayley tree. This Laplacian is used to define a discrete generalized Schrödinger operator on the tree. The case discrete fractional Schrödinger operators with index $0 < α< 2$ is considered in detail, and periodic solutions of the corresponding fractional Schrödinger equations are described. This periodicity depends on a subgroup of a group representation of the Cayley tree. For any subgroup of finite index we give a criterion for eigenvalues of the Schrödinger operator under which periodic solutions exist. For a normal subgroup of infinite index we describe a wide class of periodic solutions.

math-ph↗
arXiv2015

Self-adjointness of semi-relativistic Pauli-Fierz Hamiltonian

The spinless semi-relativistic Pauli-Fierz Hamiltonian $H$ in quantum electrodynamics is considered. The self-adjointness and essential self-adjointness of $H$ are shown. It is emphasized that it includes the massless case. Furthermore, the self-adjointness and the essential self-adjointness of the semi-relativistic Pauli-Fierz model with a fixed total momentum is also proven.

math-ph↗
arXiv2015

Translation invariant models in QFT without ultraviolet cutoffs

The translation invariant model in quantum field theory is considered by functional integrations. Ultraviolet renormalization of the translation invariant Nelson model with a fixed total momentum is proven by functional integrations. As a corollary it can be shown that the Nelson Hamiltonian with zero total momentum has a ground state for arbitrary values of coupling constants in two dimension. Furthermore the ultraviolet renormalization of the polaron model is also studied.

math-ph↗
arXiv2015

Enhanced binding of an N-particle system interacting with a scalar field II.Relativistic version

An enhanced binding of $N$-{\it relativistic} particles coupled to a massless scalar bose field is investigated. It is not assumed that the system has a ground state for the zero-coupling. It is shown, however, that there exists a ground state for sufficiently large coupling. The proof is based on checking the stability condition and showing a uniform exponential decay of infrared regularized ground states.

math-ph↗
arXiv2014

Spin-Boson Model through a Poisson-Driven Stochastic Process

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.

math-ph↗
arXiv2014

Functional integral approach to semi-relativistic Pauli-Fierz models

By means of functional integrations spectral properties of semi-relativistic Pauli-Fierz Hamiltonians in quantum electrodynamics is considered. Two self-adjoint extensions of a semi-relativistic Pauli-Fierz Hamiltonian are defined. An essential self-adjointness, a spatial decay of bound states, a Gaussian domination of the ground state and the existence of a measure associated with the ground state are shown.

math-ph↗
arXiv2014

Spectrum of the semi-relativistic Pauli-Fierz model I

HVZ type theorem for semi-relativistic Pauli-Fierz Hamiltonian, $$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$ in quantum electrodynamics is studied. Here $H$ is a self-adjoint operator in Hilbert space $\LR\otimes \fff\cong \int^\oplus_{\RR^d}\fff {\rm d}x$, and $A=\int^\oplus_{\RR^d} A(x) {\rm d}x$ a quantized radiation field and $\hf$ the free field Hamiltonian defined by the second quantization of a dispersion relation $ω:\RR^d\to \RR$. It is emphasized that massless case, $M=0$, is included. Let $E=\inf σ(\HHH)$ be the bottom of the spectrum of $\HHH$. Suppose that the infimum of $ω$ is $m>0$. Then it is shown that $σ_{\rm ess}(\HHH)=[E+m, \infty)$. In particular the existence of the ground state of $\HHH$ can be proven.

math-ph↗
arXiv2013

The Spectrum of Non-Local Discrete Schroedinger Operators with a delta-Potential

The behaviour of the spectral edges (embedded eigenvalues and resonances) is discussed at the two ends of the continuous spectrum of non-local discrete Schrödinger operators with a $δ$-potential. These operators arise by replacing the discrete Laplacian by a strictly increasing $C^1$-function of the discrete Laplacian. The dependence of the results on this function and the lattice dimension are explicitly derived. It is found that while in the case of the discrete Schrödinger operator these behaviours are the same no matter which end of the continuous spectrum is considered, an asymmetry occurs for the non-local cases. A classification with respect to the spectral edge behaviour is also offered.

math-ph↗
arXiv2012

Probabilistic Representation and Fall-Off of Bound States of Relativistic Schrödinger Operators with Spin 1/2

A Feynman-Kac type formula of relativistic Schrödinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound states is established separately for growing and decaying potentials by using martingale methods.

math-ph↗
arXiv2012

Note on the spectrum of discrete Schrödinger operators

The spectrum of discrete Schrödinger operator $L+V$ on the $d$-dimensional lattice is considered, where $L$ denotes the discrete Laplacian and $V$ a delta function with mass at a single point. Eigenvalues of $L+V$ are specified and the absence of singular continuous spectrum is proven. In particular it is shown that an embedded eigenvalue does appear for $d\geq5$ but does not for $1\leq d\leq 4$.

math-ph↗
arXiv2012

Enhanced Binding in Quantum Field Theory

This lecture note consists of three parts. Fundamental facts on Boson Fock space are introduced in Part I. Ref. 1.and 3. are reviewed in Part II and, Ref. 2. and 4. in Part III. In Part I a symplectic structure of a Boson Fock space is studied and a projective unitary representation of an infinite dimensional symplectic group through Bogoliubov transformations is constructed. In Part II the so-called Pauli-Fierz model (PF model) with the dipole approximation in non-relativistic quantum electrodynamics is investigated. This model describes a minimal interaction between a massless quantized radiation field and a quantum mechanical particle (electron) governed by Schrödinger operator. By applying the Bogoliubov transformation introduced in Part I we investigate the spectrum of the PF model. First the translation invariant case is considered and the dressed electron state with a fixed momentum is studied. Secondly the absence of ground state is proven by extending the Birman-Schwinger principle. Finally the enhanced binding of a ground state is discussed and the transition from unbinding to binding is shown. In Part III the so-called $N$-body Nelson model is studied. This model describes a linear interaction between a scalar field and $N$-body quantum mechanical particles. First the enhanced binding is shown by checking the so-called stability condition. Secondly the Nelson model with variable coefficients is discussed, which model can be derived when the Minkowskian space-time is replaced by a static Riemannian manifold, and the absence of ground state is proven, if the variable mass decays to zero sufficiently fast. The strategy is based on a path measure argument.

math-ph↗