Masao Hirokawa

Ground States of a General Class of Quantum Field Hamiltonians

We consider a model of a quantum mechanical system coupled to a (massless) Bose field, called the generalized spin-boson model (A. Arai and M. Hirokawa, J. Funct. Anal. 151 (1997), 455–503), without infrared regularity condition. We define a regularized Hamiltonian H (ν) with a parameter ν ≥ 0 such that H = H (0) is the Hamiltonian of the original model. We clarify a relation between ground states of H (ν) and those of H by formulating sufficient conditions under which weak limits, as ν → 0, of the ground states of H (ν)s are those of H. We also establish existence theorems on ground states of H (ν) and H under weaker conditions than in the previous paper mentioned above.

REMARKS ON THE GROUND STATE ENERGY OF THE SPIN-BOSON MODEL: AN APPLICATION OF THE WIGNER–WEISSKOPF MODEL

For the ground state energy of the spin-boson (SB) model, we give a new upper bound in the case with infrared singularity condition (i.e. without infrared cutoff), and a new lower bound in the case of massless bosons with infrared regularity condition. We first investigate spectral properties of the Wigner–Weisskopf (WW) model, and apply them to SB model to achieve our purpose. Then, as an extra result of the spectral analysis for WW model, we show that a non-perturbative ground state appears, and its ground state energy is so low that we cannot conjecture it by using the regular perturbation theory.

Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field

It is investigated that the structure of the kernel of the Dirac–Weyl operator D of a charged particle in the magnetic-field B=B0+B1, given by the sum of a strongly singular magnetic field B0(⋅)=Σνγνδ(⋅−aν) with some singular points aν and a magnetic-field B1 with a bounded support. Here the magnetic field B1 may have some singular points with the order of the singularity less than 2. At a glance, it seems that, following “Aharonov–Casher Theorem” [Phys. Rev. A 19, 2461 (1979)], the dimension of the kernel of D, dim ker D, is a function of one variable of the total magnetic flux (=Σνγν+∫R2B1dxdy) of B. However, since the influence of the strongly singular points works, dim ker D indeed is a function of several variables of the total magnetic flux and each of γν’s.

Role of a Phase Factor in the Boundary Condition of a One-Dimensional Junction

One-dimensional quantum systems can be experimentally studied in recent nanotechnology like the carbon nanotube and the nanowire. We have considered the mathematical model of a one-dimensional Schrodinger particle with a junction and have analyzed the phase factor in the boundary condition of the junction. We have shown that the phase factor in the tunneling case appears in the situation of the non-adiabatic transition with the three energy levels in the exact Wentzel–Kramers–Brillouin analysis.

Dicke-type energy level crossings in cavity-induced atom cooling: Another superradiant cooling

This paper is devoted to energy-spectral analysis for the system of a two-level atom coupled with photons in a cavity. It is shown that the Dicke-type energy level crossings take place when the atom-cavity interaction of the system undergoes changes between the weak coupling regime and the strong one. Using the phenomenon of the crossings we develop the idea of cavity-induced atom cooling proposed by the group of Ritsch, and we lay mathematical foundations of a possible mechanism for another superradiant cooling in addition to that proposed by Domokos and Ritsch. The process of our superradiant cooling can function well by cavity decay and by control of the position of the atom, at least in (mathematical) theory, even if there is neither atomic absorption nor atomic emission of photons.

Ground state transition for two-level system coupled with Bose field

For the two-level system coupled with a Bose field, we investigate the ground state transition and appearance of a non-perturbative ground state.

Boundary Conditions in One-dimensional Tunneling Junction

The tunneling effect is first characterized in quantum mechanics. However, the mathematical analysis for the tunneling effect is so complicated. In the conventional sense, the WKB analysis is useful to analyze such effect. Here, we consider the mathematical analysis for the tunneling effect in the one-dimensional Schr?dinger system from the viewpoint of functional analysis. From the rigorous analysis, we obtain the interference pattern of the tunneling particle.

One-Dimensional Tunnel-Junction Formula for the Schrödinger Particle

We handle all the self-adjoint extensions of the minimal Schrodinger operator for the nonrelativistic electron living in the one-dimensional configuration space with a junction. We are interested in every boundary condition corresponding to the individual self-adjoint extension. Thus, we clarify all the types of those boundary conditions of the wave functions of the nonrelativistic electron. We find concrete tunnel-junction formulae for the nonrelativistic electron passing through the junction, which reveals the phase factor caused by the tunneling through the junction. Using this tunnel-junction formula, we propose a mathematical possibility of a tunnel-junction device for qubit.

Stability of Ground States in Sectors and Its Application to the Wigner-Weisskopf Model

We consider two kinds of stability (under a perturbation) of the ground state of a self-adjoint operator: the one is concerned with the sector to which the ground state belongs and the other is about the uniqueness of the ground state. As an application to the Wigner–Weisskopf model which describes one mode fermion coupled to a quantum scalar field, we prove in the massive case the following: (a) For a value of the coupling constant, the Wigner–Weisskopf model has degenerate ground states; (b) for a value of the coupling constant, the Wigner–Weisskopf model has a first excited state with energy level below the bottom of the essential spectrum. These phenomena are nonperturbative.

Anomalous Pauli Electron States for Magnetic Fields with Tails

AbstractWe consider a two-dimensional electron with an anomalous magnetic moment, g>2, interacting with a nonzero magnetic field B perpendicular to the plane which gives rise to a flux F. Recent results about the discrete spectrum of the Pauli operator are extended to fields with the