Mathematical Analysis of a Generalized Chiral Quark Soliton Model

A generalized version of the so-called chiral quark soliton model (CQSM) in nuclear physics is introduced. The Hamiltonian of the generalized CQSM is given by a Dirac type operator with a mass term being an operator-valued function. Some mathematically rigorous results on the model are reported. The subjects included are: (i) supersymmetric structure; (ii) spectral properties; (iii) symmetry reduction; (iv) a unitarily equivalent model.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

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.Comment: We add Sections 1.1,1.2 and 1.

The Lamb Shift from an Effective Hamiltonian in Non-relativistic Quantum Electrodynamics and a General Class of Effective Operators

Some aspects of spectral analysis of an effective Hamiltonian in non-relativistic quantum electrodynamics are reviewed. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian. Moreover, a general class of effective operators is presented, which comes from models of an abstract quantum system interacting with a Bose field

A criterion for the boundedness from below with a class of symmetric operators and its applications

AbstractA class of symmetric operators H acting in L2(M, μ) with an abstract measure space < M,μ > is considered and a criterion for the boundedness from below of H is presented. The criterion gives also a method to estimate the lower bound of H. As a special case of the criterion, an affirmative answer is given to the question: “If an eigenfunction of a self-adjoint operator in L2(M,μ) is strictly positive or “nodeless,” then is the corresponding eigenvalue the lowest point of the spectrum?” The class includes Schrödinger operators on Rn with Rn M=Rnand μ being the Lebesgue measure. It is illustrated how the method may give a simple proof of the boundedness from below for some Schrödinger operators. Further, it is shown (1) that, in a special case, there exists an operator in L2(M, μ) essentially isospectral to H and (2) that the existence of a “nodeless” eigenfunction of H implies the existence of a supersymmetric quantum mechanics in which H—λ is unitarily equivalent to the restriction of the supersymmetric Hamiltonian to a subspace, where λ is the lower bound of H

Derivation of the Lamb Shift from an Effective Hamiltonian in Non-relativistic Quantum Electrodynamics

Some aspects of spectral analysis of an effective Hamiltonian in non-relativistic quantum electrodynamics are reviewed. The Lamb shift of a hydrogen-like atom is derived as the lowest order approximation (in the fine structure constant) of an energy level shift of the effective Hamiltonian. Moreover, a general class of effective operators is presented, which comes from models of an abstract quantum system interacting with a Bose field

On Arithmetic Quantum Field Theory

This is the proceedings of the 2nd Japanese-German Symposium on Infinite Dimensional Harmonic Analysis held from September 20th to September 24th 1999 at the Department of Mathematics of Kyoto University.この論文集は, 1999年9月20日から9月24日の日程で京都大学理学研究科数学教室において開催された第2回日独セミナー「無限次元調和解祈」の成果をもとに編集されたものである.編集 : ハーバート・ハイヤー, 平井 武, 尾畑 信明Editors: Herbert Heyer, Takeshi Hirai, Nobuaki Obata #enWe review fundamental aspects of arithmetic quantum field theory

Inequivalence of Quantum Dirac Fields of Different Masses and General Structures Behind

A family of irreducible representations of the canonical anticommutation rela- tions (CAR) over an abstract Hilbert space indexed by a set of bounded linear operators is presented and a theorem on the mutual equivalence of them is estab- lished. As an application of the theorem, it is proved that quantum Dirac elds of different masses are mutually inequivalent. Moreover, a new class of irreducible representations of the CAR over a Hilbert space, which includes, as a special case, time-zero quantum Dirac elds, is constructed