Volume 41, Issue 7, July 2000
 QUANTUM PHYSICS; PARTICLES AND FIELDS


On a multidimensional Schrödinger–Poisson scattering model for semiconductors
View Description Hide DescriptionWe consider a stationary Schrödinger–Poisson problem modeling a selfconsistent transport in a quantum coupler. The Schrödinger equation is set on a bounded domain with transparent boundary conditions describing incoming scattering states of the Schrödinger operator. The coupling with the Poisson equation is done thanks to a nonlinear limiting absorption procedure. The charge density of the limit potential is shown to be equal to the sum of the scattering state and bound state densities.

Factorization and smallenergy asymptotics for the radial Schrödinger equation
View Description Hide DescriptionThe radial Schrödinger equation is considered when the potential is real valued, is integrable, and has a finite first moment. The Jost function, the scattering matrix, the number of bound states for the potential are expressed in terms of the corresponding quantities associated with the fragments of the potential. An improved expansion on the smallenergy asymptotics of the Jost solution is presented.

A particlefield Hamiltonian in relativistic quantum electrodynamics
View Description Hide DescriptionWe mathematically analyze a Hamiltonian of a Dirac particle—a relativistic charged particle with spin 1/2—minimally coupled to the quantized radiation field, acting in the Hilbert space where is the Fock space of the quantized radiation field in the Coulomb gauge, is an external potential in which the Dirac particle moves, is a photonmomentum cutoff function in the interaction between the Dirac particle and the quantized radiation field, and is a deformation parameter connecting the Hamiltonian with the “dipole approximation” and the original Hamiltonian We first discuss the selfadjointness problem of Then we consider the Hamiltonian without the external potential. It is shown that, under a general condition on the closure of is unitarily equivalent to a direct integral with a fiber Hamiltonian acting in the four direct sum of physically the polaron Hamiltonian of the Dirac particle with total momentum

Companion equations for branes
View Description Hide DescriptionThe quantum mechanical transition between a free particle Lagrangian and the Klein–Gordon field description of a free particle (particlewave duality) is conjectured to extend to an analogous construction of relativistically invariant wave equations associated with strings and branes.Electromagnetic interactions in the two systems are discussed. It is emphasized that all integrable free field theories, including those of Dirac–Born–Infeld type, are associated with Lagrangians equivalent to divergences on the space of solutions of the equations of motion.

On the time evolution of the meanfield polaron
View Description Hide DescriptionIn this paper a meanfieldtheory for the evolution of an electron in a crystal is proposed in the framework of the Schrödinger formalism. The wellposedness of the problem as well as the conservation laws associated to the invariances of the Action Functional of the problem and the stability of the minimal energy solution are studied.

Abelian theory and spherically symmetric electromagnetism
View Description Hide DescriptionThree different methods to quantize the spherically symmetric sector of electromagnetism are presented: First, it is shown that this sector is equivalent to Abelian theory in four spacetime dimensions with suitable boundary conditions. This theory, in turn, is quantized by both a reduced phase space quantization and a spin network quantization. Finally, the outcome is compared with the results obtained in the recently proposed general quantum symmetry reduction scheme. In the magnetically uncharged sector, where all three approaches apply, they all lead to the same quantum theory.

Probability amplitude dynamics for a twolevel system
View Description Hide DescriptionThe timedependent probability amplitudes are determined for a twolevel system without invoking the rotating wave approximation. A new analytic solution is obtained in the limit of small ratio of Rabi frequency to driver frequency. An analytic solution is also obtained in the limit of large ratio of Rabi frequency to driver frequency for a restricted range of parameters. The form of this solution guides the selection of parameters that cause substantial changes in the character of the solution. The dependence on parameter values is studied numerically for the transition probability and the coherent spectrum.

From unsharp to sharp quantum observables: The general Hilbert space case
View Description Hide DescriptionAn unsharp quantum observable can be considered a realization of a sharp observable if and only if it is commutative. In this paper we describe an explicit procedure for reconstructing such a sharp observable and for establishing the probabilistic correlations between the sharp reconstruction and the given unsharp observable.

Topological tensor current of branes in the φmapping theory
View Description Hide DescriptionWe present a new topological tensorcurrent of branes by making use of the φmapping theory. It is shown that the current is identically conserved and behaves as and every isolated zero of the vector field corresponds to a “magnetic” brane. Using this topological current, the generalized Nambu action for multi branes is given, and the field strength corresponding to this topological tensorcurrent is obtained. It is also shown that the magnetic charges carried by branes are topologically quantized and labeled by Hopf index and Brouwer degree, the winding number of the φ mapping.

Energy in Yang–Mills on a Riemann surface
View Description Hide DescriptionSengupta’s lower bound for the Yang–Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang–Mills critical sets correspond to critical sets of the energy action on a space of paths. This may shed light on a conjecture of Atiyah and Bott concerning Morse theory for A/G.

Resonances for laterally coupled quantum waveguides
View Description Hide DescriptionA system of two waveguides coupled laterally through a small window is considered. The asymptotics (in the width of window) of resonance (quasibound state) close to the second threshold is obtained. The cases of two different and two identical waveguides are considered. The technique is matching the asymptotic expansions of the solutions.

Note on coherent states and adiabatic connections, curvatures
View Description Hide DescriptionWe give a possible generalization to the example in the paper of Zanardi and Rasetti [Phys. Lett. A 264, 94 (1999)]. For this, explicit forms of adiabatic connection, curvature, etc., are given. We also discuss the possibility of another generalization of their model.

The analogue of the chain with a boundary
View Description Hide DescriptionWe study the analogue of the spin chain with a boundary magnetic field We construct explicit bosonic formulas of the vacuum vector and the dual vacuum vector with a boundary magnetic field. For an application of the explicit formula of the vacuum vector, we derive the integral representation of the boundary spontaneous magnetization.

Asymptotic dynamics in quantum field theory
View Description Hide DescriptionA crucial element of scattering theory and the LSZ reduction formula is the assumption that the coupling vanishes at large times. This is known not to hold for the theories of the Standard Model and in general such asymptotic dynamics is not well understood. We give a description of asymptotic dynamics in field theories which incorporates the important features of weak convergence and physical boundary conditions. Applications to theories with three and four point interactions are presented and the results are shown to be completely consistent with the results of perturbation theory.

Wave operators for the surface Maryland model
View Description Hide DescriptionWe study scatteringproperties of the discrete Laplacian H on the halfspace with the boundary condition where We denote by the Dirichlet Laplacian on Khoruzenko and Pastur [Phys. Rep. 288, 109–126 (1997)] have shown that if α has typical Diophantine properties then the spectrum of H on is pure point and that corresponding eigenfunctions decay exponentially. We demonstrated in an earlier paper [Lett. Math. Phys. 45, 185 (1998)] that for every α independent over the rationals the spectrum of H on is purely absolutely continuous. In this paper, we continue the analysis of H on and prove that whenever α is independent over the rationals, the wave operators exist and are complete on Moreover, we show that under the same conditions H has no surface states on

A superspace formulation of an “asymptotic” OSp(3,12) invariance of Yang–Mills theories
View Description Hide DescriptionWe formulate a superspace field theory which is shown to be equivalent to the symmetric BRS/antiBRS invariant Yang–Mills action. The theory uses a sixdimensional superspace and one OSp(3,12) vector multiplet of unconstrained superfields. We establish a superspace WT identity and show that the formulation has an asymptotic OSp(3,12) invariance as the gauge parameter goes to infinity. We give a physical interpretation of this asymptotic OSp(3,12) invariance as a symmetry transformation among the longitudinal/time like degrees of freedom of and the ghost degrees of freedom.

Phase space tunneling for operators with symbols in a Gevrey class
View Description Hide DescriptionPhase space tunneling and exponential decay of eigenfunctions in phase space are well known for operators with symbols which are analytic in some neighborhood of the real axis. This can be used to prove an adiabatic theorem of exponential order if one assumes the Hamiltonian to depend analytically on time. However to study compactly supported switching processes one has to weaken the analyticity assumptions. Here we examine nonanalytic symbols with Gevrey class regularity and show that we get an exponential decay of the corresponding eigenfunctions with respect to as where The loss of regularity causes a slower decay in The analysis is done using the methods of Martinez and its generalization by Nakamura. An upper bound for the rate of decay is given.

Time ordering, energy ordering, and factorization
View Description Hide DescriptionRelations between integrals of timeordered product of operators, and their representation in terms of energyordered products are studied. Both can be decomposed into irreducible factors and these relations are discussed as well. The energyordered representation was invented to separate various infrared contributions in gauge theories. It is shown that the irreducible timeordered expressions can be used to accomplish the same purpose. Besides, it has the added advantage of being factorizable.

Quantum grammars
View Description Hide DescriptionWe consider quantum (unitary) continuous time evolution of spins on a lattice together with quantum evolution of the lattice itself. In physics such evolution was discussed in connection with quantum gravity. It is also related to what is called quantum circuits, one of the incarnations of a quantum computer. We consider simpler models for which one can obtain exact mathematical results. We prove existence of the dynamics in both Schrödinger and Heisenberg pictures, construct KMS states on appropriate algebras. We show (for high temperatures) that for each system where the lattice undergoes quantum evolution, there is a natural scaling leading to a quantum spin system on a fixed lattice Z, defined by a renormalized Hamiltonian.

Spectral zeta functions for a cylinder and a circle
View Description Hide DescriptionSpectral zeta functions for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane containing the closed interval of real axis Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced.
