Scattering theory for a model of an atom in a quantized field

We compute the scattering operator for a simplified model of an atom interacting a with a quantized field. The field is restricted to the vacuum and one-particle sectors, and the atom has only two states. We also solve the inverse scattering problem for the same model. The methods used rely on the particular form of the interaction, which is chosen to mimic the interaction between radiation and matter.This work was partially supported by NSF Grant DMS-8922941.     

Dynamical R-matrices for Calogero models

We present a systematic study of the integrability of the Calogero models, degenerate as well as elliptic, associated with arbitrary (semi-)simple Lie algebras and with symmetric pairs of Lie algebras, where “integrability” is understood to encompass not only the existence of a Lax representation for the equations of motion but also the—more far-reaching—existence of a (dynamical) R-matrix. Using the standard group-theoretical machinery available in this context, we show that integrability of these models, in this sense, can be reduced to the existence of a certain function, denoted here by F, defined on the relevant root system and taking values in the respective Cartan subalgebra, subject to a rather simple set of algebraic constraints: these ensure, in one stroke, the existence of a Lax representation and of a dynamical R-matrix, all given by explicit formulas. We also show that among the simple Lie algebras, only those belonging to the A-series admit a solution of these constraints, whereas the AIII-series of symmetric pairs of Lie algebras, corresponding to the complex Grassmannians SU(p,q)/S(U(p)×U(q)), allows non-trivial solutions when |p−q|⩽1. Apart from reproducing all presently known dynamical R-matrices for Calogero models, our method provides new ones, namely for the degenerate models when |p−q|=1 and for the elliptic models when |p−q|=1 or p=q.     

Exponential decay near resonance,without analyticity

We consider a quantum mechanical model which displays the behaviour associated with having a resonance or metastable state. The Hamiltonian depends on a parameter . When =0, there is an eigenstate ₀; when 0, ₀ dissolves into the continuous spectrum, showing approximate exponential decay. We prove this result without using dilatation analyticity. The model describes a two-state atom coupled to the quantized radiation field. The state space of the field is truncated, so that only the vacuum and one-photon states are included.This work was partially supported by NSF Grant DMS-8922941     

SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

Existence theorems for π/n vortex scattering

The analysis of 90° vortex-vortex scattering is extended to /n scattering in all head-on collisions ofn vortices in the Abelian Higgs model. A Cauchy problem with initial data that describe the scattering ofn vortices is formulated. It is shown that this Cauchy problem has a unique global finite-energy solution. The symmetry of the solution and the form of the local analytic solution then show that /n scattering is realized.     

Calogero-Moser models with noncommutative spin interactions

We construct integrable generalizations of the elliptic Calogero-Sutherland-Moser model of particles with spin, involving noncommutative spin interactions. The spin coupling potential is a modular function and, generically, breaks the global spin symmetry of the model down to a product of U(1) phase symmetries. Previously known models are recovered as special cases.     

On the solutions of 3-particle spin elliptic Calogero-Moser systems

We describe a class of the singular solutions to the multicomponent analogs of the Lamé equation, arising as equations of motion of the elliptic Calogero-Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.     

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Poincaré-invariant generalizations of the Galilei-invariant Calogero-MoserN-particle systems are studied. A quantization of the classical integralsS ₁, ...,S _N is presented such that the operators ₁, ..., _N mutually commute. As a corollary it follows thatS ₁, ...,S _N Poisson commute. These results hinge on functional equations satisfied by the Weierstrass - and -functions. A generalized Cauchy identity involving the -function leads to anN×N matrixL whose symmetric functions are proportional toS ₁, ...,S _N .     

Nonexistence theorems for Yang-Mills fields outside the black hole of the Schwarzschild spacetime

In this Letter, the authors prove some nonexistence theorems for regular static Yang-Mills fields outside the black hole of the Schwarzschild spacetime under certain boundary conditions.Research work supported by the International Centre for Theoretical Physics and the Science Fund of the Chinese Academy of Sciences.     

On the finiteness of the discrete spectrum of Hamiltonians for quantum systems of three one- or two-dimensional particles

The system of three particles (acting in K dimensions (K=1, 2)) without stable two-particle subsystems is studied. For short-range potentials V _ij (r _ij ), the finiteness of the discrete spectrum is proved for any type of permutational symmetry.     

Trace identities from tau functions

An approach to trace identities is described based on the-function formalism for mKdV and affine Toda systems. The set-up is given for discrete and continuous spectrum solutions by exploiting the relation between Jost solutions, diagonal entries of the monodromy matrix, and-functions. Examples for solutions with discrete scattering data are worked out.     

Scattering on the hyperbolic plane in the Aharonov-Bohm gauge field

The scattering of a quantum charged particle on the hyperbolic plane in the Abelian Aharonov-Bohm field is considered. S-matrix, scattering amplitude and its flat space limit are constructed in a closed form by the Sommerfeld integral approach.     

Schwinger's method for the massive Casimir effect

We apply a method recently proposed by Schwinger to the massive scalar field to calculate the Casimir effect. The method is applied with two different regularization schemes: Schwinger's original one by means of Poisson formula and another one by means of analytical continuation.     

Schwinger's method for the fermionic Casimir effect

The Casimir energy of a massive Dirac field confined between two parallel infinite plates is computed using a method proposed by Schwinger. The massless case is obtained as a limit of the massive case. The boundary conditions are those of zero current through the plates, as inspired by quark confinement in the MIT bag model for hadrons. We use an analytical continuation method of regularization which allows the employment of Epstein function techniques. The calculation using Schwinger's original regularization by a cutoff in proper time is also outlined.     

On Schwinger's method for obtaining the Casimir effect

We show that a slight modification in Schwinger's method leads directly to the finite result for the Casimir energy of a massless scalar theory.     

The inverse scattering matrix for the Schrödinger equation when the potentialq(x) ∈L 1 1 with a singular term

When the potentialq(x) L ₁ ¹ with a singular term, the continuities of the scattering matrix of the Schrödinger equation are investigated. By means of the transformation approach, we arrive at the conclusion that the scattering matrix S(k) of such a potential is continuous for the wholek,- <k < .     

Schwinger's method for the fermionic Casimir effect at finite temperature

The Casimir energy of a massive Dirac field at finite temperature and confined between two parallel infinite plates is computed using a method proposed by Schwinger. The boundary conditions are those of zero current through the plates, as inspired by quark confinement in the MIT bag model for hadrons. We use Schwinger's original regularization by a cutoff in proper-time. We comment on the analytical continuation method of regularization, which allows the employment of Epstein function techniques, and on the different possible expressions for the final answer.