General Decomposition of Radial Functions on Rn and Applications to N-Body Quantum Systems

We present a generalization of the Fefferman–de la Llave decomposition of the Coulomb potential to quite arbitrary radial functions V on Rⁿ going to zero at infinity. This generalized decomposition can be used to extend previous results on N-body quantum systems with Coulomb interaction to a more general class of interactions. As an example of such an application, we derive the high density asymptotics of the ground state energy of jellium with Yukawa interaction in the thermodynamic limit, using a correlation estimate by Graf and Solovej.     

A Dynamical Mapping Method in Nonrelativistic Models of Quantum Field Theory

Abstract

Exact solutions of Heisenberg equations and two-particle eigenvalue problems for the nonrelativistic four-fermion interaction and N, Θ model are obtained in the framework of a dynamical mapping method. Equivalence of different dynamical mappings is shown.     

Scattering Matrix in Quantum Field Theory

We show that a relativistically covariant, unitary, and causative scattering matrix, which is finite in each order of perturbation theory, can be constructed using fundamental postulates of quantum field theory and give a physical interpretation of the results obtained.     

Mathematical Scattering Theory in Quantum Waveguides

Doklady Physics - A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form $$i{{partial }_{t}}f = mathcal{A}f$$, where...     

共振光散射的时间量子理论

本文发展了Ｓｃｈｏｍｍｅｒｓ的时间观点，定义了时间，尤其是表征量子系统光散射的散射时间和共振散射时间。通过散射时间本征态的假定实现了散射时间的量子化，得到了量子化的共振散射时间。对原子的弹性光散射和Ｒａｍａｎ散射的成功应用推出了原子和原子的价电子逐级电离所形成的离子的所有原子能级的普适近似公式。     

Symmetrized Wavefunctions in a Quantum N-Body System

A general method for symmetrizing the generalized radial functions in a quantum N-body system to belong to the irreducible representation of the proper point symmetry group of the system is presented, based on irreducible bases in the point group space.     

Spin Correlations in Nonrelativistic Quantum Mechanics

Einstein—Podolsky—Rosen spin correlations in the framework of nonrelativistic quantum mechanics for moving observers are calculated. The measurements are performed in bounded regions of space (detectors), not necessarily simultaneously. The resulting correlation function depends not only on the directions of spin measurements but also on the relative velocity of the observers.     

Nonrelativistic Quantum Mechanics with Fundamental Environment

Spontaneous transitions between bound states of an atomic system, “Lamb Shift” of energy levels and many other phenomena in real nonrelativistic quantum systems are connected within the influence of the quantum vacuum fluctuations (fundamental environment (FE)) which are impossible to consider in the limits of standard quantum-mechanical approaches. The joint system “quantum system (QS) + FE” is described in the framework of the stochastic differential equation (SDE) of Langevin-Schr?dinger (L-Sch) type, and is defined on the extended space R ³ ⊗ R _{ξ}, where R ³ and R _{ξ} are the Euclidean and functional spaces, respectively. The density matrix for single QS in FE is defined. The entropy of QS entangled with FE is defined and investigated in detail. It is proved that as a result of interaction of QS with environment there arise structures of various topologies which are a new quantum property of the system.     

广义基的多重散射理论

提出了任意函数基表象中的多重散射理论. 基于这个框架, 一组新基的多重散射计算表明其收敛性比传统的球谐函数基的要好, 从而可以实现更快和更节省物理内存的计算. 另外, 将多重散射矩阵分为传播矩阵和散射矩阵, 很好地描述了散射性质和减小了多重散射计算中块矩阵的尺寸.     

Radiation Reaction of a Nonrelativistic Quantum Charged Particle

An alternative approach to analyze the nonrelativistic quantum dynamics of a rigid and extended charged particle taking into account the radiation reaction is discussed with detail. Interpretation of the field operators as annihilation and creation ones, theory of perturbations and renormalization are not used. The analysis is carried out in the Heisenberg picture with the electromagnetic field expanded in a complete orthogonal basis set of functions which allows the electromagnetic field to satisfy arbitrary boundary conditions. The corresponding coefficients are the field operators which satisfy the usual commutation relations. A nonlinear equation of motion for the charged particle is obtained. A careful consideration of the quantum effects allows the derivation of a linear equation of motion which is free of both runaway solutions and preacceleration, even for a point charge. Also, the electromagnetic mass, which is defined as the coefficient of the acceleration operator, vanishes for a point particle. However, this does not mean that the results are free of ambiguities which are exhibited and discussed.     

The Phase Space Model of Nonrelativistic Quantum Mechanics

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.     

Asymptotic Completeness for a Renormalized Nonrelativistic Hamiltonian in Quantum Field Theory: The Nelson Model

Scattering theory for the Nelson model is studied. We show Rosen estimates and we prove the existence of a ground state for the Nelson Hamiltonian. Also we prove that it has a locally finite pure point spectrum outside its thresholds. We study the asymptotic fields and the existence of the wave operators. Finally we show asymptotic completeness for the Nelson Hamiltonian.     

The Quantum N-Body Problem and the Auxiliary Field Method

Approximate analytical energy formulas for N-body semirelativistic Hamiltonians with one- and two-body interactions are obtained within the framework of the auxiliary field method. We first review the method in the case of nonrelativistic two-body problems. A general procedure is then given for N-body systems and applied to the case of baryons in the large-N _c limit.     

General Requirements for a Relativistic Quantum Theory

The equations of a relativistic quantum theory for two or more particles should satisfy at least the following criteria. (1) They should be Poincaré invariant. (2) The cluster property should hold. (3) Causality should not be violated over distances much larger than the Compton wavelengths of the particles involved. (4) The electromagnetic interaction between charged particles should be formulated in a gauge-invariant way. (5) If, for a two-particle system, one of the masses becomes infinitely large, the equations should reduce to the relevant relativistic equation for the other particle. (6) In the nonrelativistic limit the equation should reduce to the Schr?dinger equation. In this paper it will be shown how a quasi-potential theory, which was introduced many years ago [1] and which was applied to a number of systems [2–12], meets all these requirements. Received March 25, 1997; revised July 15, 1997; accepted for publication March 18, 1998     

Scattering Theory for Open Quantum Systems with Finite Rank Coupling

Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator A _D in a Hilbert space \({\mathfrak H}\) is used to describe an open quantum system. In this case the minimal self-adjoint dilation \(\widetilde K\) of A _D can be regarded as the Hamiltonian of a closed system which contains the open system \(\{A_{\!D},{\mathfrak H}\}\), but since \(\widetilde K\) is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family {A(μ)} of maximal dissipative operators depending on energy μ, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm–Liouville operators arising in dissipative and quantum transmitting Schrödinger–Poisson systems.     

Exact Path-Integral Representations for the T-Matrix in Nonrelativistic Potential Scattering

Several path integral representations for the T-matrix in nonrelativistic potential scattering are given which produce the complete Born series when expanded to all orders and the eikonal approximation if the quantum fluctuations are suppressed. They are obtained with the help of ??phantom?? degrees of freedom which take away explicit phases that diverge for asymptotic times. Energy conservation is enforced by imposing a Faddeev?CPopov-like constraint in the velocity path integral. An attempt is made to evaluate stochastically the real-time path integral for potential scattering and generalizations to relativistic scattering are discussed.     

Instantaneous Spreading Versus Space Localization for Nonrelativistic Quantum Systems

A theorem of Hegerfeldt (Kielanowski et al. 1998) establishes, for a class of quantum systems, a dichotomy between those which are permanently localized in a bounded region of space, and those exhibiting instantaneous spreading. We analyze in some detail the physical inconsistencies which follow from both of these options, and formulate which, in our view, are the basic open problems.