Bengt Nagel

The relativistic Hermite polynomial is a Gegenbauer polynomial

It is shown that the polynomials introduced recently by Aldaya, Bisquert, and Navarro‐Salas [Phys. Lett. A 156, 381 (1991)] in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of Gegenbauer polynomials. This fact is useful in the investigation of the properties of the corresponding wave function. Some examples are given, in particular, related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers.

A generalized edge of the wedge theorem

The edge of the wedge theorem is generalized to the case where one only assumes the existence and equality of the distribution boundary values off±(z) and all their derivatives on some analytic curveC inRn. Heref±(z) are holomorphic inRn±iC, respectively, whereC is a convex cone, andC has its tangent vector inC at every point. Under these assumptions there exists an analytic continuationf(z) holomorphic in some complex neighbourhood of the double cone generated byC. A proof is also given of the connection between the existence of a distribution boundary value and the growth of the holomorphic function near the boundary.

Some results in non-commutative ergodic theory

We study some properties of invariant states on aC*-algebraA with a groupG of automorphisms. Using the concept ofG-factorial state, which is a “non-commutative” generalization of the concept of ergodic measure, in general wider in scope thanG-ergodic state, we show that under a certain abelianity condition on (A,G), which in particular holds for the quasi-local algebras used in statistical mechanics, two differentG-ergodic states are disjoint. We also define the concept ofG-factorial linear functional, and show that under the same abelianity condition such a functional is proportional to aG-ergodic state. This generalizes an earlier result for complex ergodic measures.

SPECTRA AND GENERALIZED EIGENFUNCTIONS OF THE ONE- AND TWO­ MODE SQUEEZING OPERATORS IN QUANTUM OPTICS

In recent years the concept of squeezed state has become central in quantum optics both from the theoretical and experimental point of view [1]. In the simplest one-mode case a squeezed state is defined here as a displaced squeezed vacuum of the form |α, ς) = D(α)S(ς)|0> obtained by applying the squeezing operator S(ς) = exp[(ς *a 2 — ς a +2 ) 12] [2] and the displacement operator D(α) = exp( α a + — α * a) on the vacuum state |0> (i e the ground state of the one-dimensional harmonic oscillator), ς = re 12θ and α = |α| e is are complex numbers, and a = (q + ip)/√2, a + =(q-ip)/ √ 2 in terms of the normalized coordinate and momentum operators q and p , where [q, p] = iI The term “squeezing” comes from the fact that for cos 2θ > tanh r (> 0) the dispersion Δq of the coordinate operator in the squeezed state is smaller than the vacuum state value 1/√2 : the uncertainty of the value of q is squeezed compared to the vacuum value. If one performs a rotation in the qp-plane (“phase space”) to new canonically conjugate operators q θ = cos θ q + sin θ p , p θ = sin θ q + cos θ p the dispersions Δ q θ and Δ p θ will be equal to e -r / √2 and e r √2, respectively, and the correlation Δ (q θ p θ ) [3] is zero. This implies that the variance matrix of the original set qp (a symmetric 2x2 matrix with the variances (Δq)2 and ( Δ p) 2 as diagonal elements and Δ (qp) as off diagonal elements) has its determinant equal to the minimum allowed value 1/4 . This can actually be taken as a characteristic property of squeezed states: they are the states (amongst a priori even mixed states) that give equality in the relation ( Δ q) 2 ( Δ p) 2 — [ Δ (qp)] 2 ≥ 1 / 4 (sometimes called the Schrodinger-Robertson uncertainty relation).

Differentiable vectors and sharp momentum states of helicity representations of the Poincaré group

This paper discusses some mathematical difficulties in handling sharp momentum eigenvectors for a massless helicity representation of the Poincare group, related to the non‐nuclearity of the space of differentiable vectors, and to the existence of singularities in the Lorentz group generators. A simple characterization of the nuclear space of differentiable vectors of the extension of the representation to a representation of the conformal group is given in terms of functions on the space R4− {0}. Using a fibration of this space over the forward light cone (in momentum space), the singularity in the generators is shown to be related to the fact that the standard presentation of the helicity representations should be reformulated in terms of nontrivial sections over the light cone. The problem is partly identical with the one encountered in the study of an electron in a magnetic monopole field, the generator singularity taking the place of the Dirac string.

An expansion of the hypergeometric function in Bessel functions

An expansion of the hypergeometric function 2F1(a,b,c+1;−z2/4ab) in Bessel functions of argument z is derived. This expansion can be used to obtain an asymptotic expansion of the hypergeometric function for large absolute values of a and b.

Introduction to Rigged Hilbert Spaces (RHS)

The purpose of this talk is to give a simple introduction to ideas related to the concept of Rigged Hilbert Space (or Gelfand Triplet). This concept has sometimes been used in defining resonances, but time does not permit us to go into this application. However, some references are given.

Confluence expansions of the generalized hypergeometric function

By confluencing a subset of upper and lower parameters in the generalized hypergeometric function F-P(Q)(a(1,),...,a(P),c(1),...,c(Q);z) with the variable z one obtains a lower-order hypergeometric ...

Analytic vectors for the Poincaré group in quantum field theory

We give a new proof of the theorem stating that in a quantum field theory with tempered field operators the dense domain of the polynomial algebra of these field operators applied to the vacuum state contains a dense invariant set of analytic vectors for the representation of the Poincare group.

Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic, space-independent coefficients, with an application to electrodiffusion

SummaryWe study a Markovian process, the state space of which is the product of a set ofn points and the realx-axids. Under certain regularity conditions this study is equivalent to investigating the solution of a set of couple diffusion equations, generalization of the Fokker-Planck (or second Kolmogorov) equation. Assuming the process homogeneous inx, but in general time-inhomogeneous, this set of equations is studied with the help of the Fourier transformation. The marginal distribution in then discrete states corresponds to a time-inhomogeneousn-state Markov chain in continuous time. The properties of such a Markov chain are studied, especially the asymptotic behaviour in the time-periodic case. We obtain a natural generalization of the well-known asymptotic behaviour in the time-homogeneous case, finding a subdivision of the states into groups of essential states, the distribution inside easch group being asymptotically periodic and independent of the starting distribution. Next, still assuming time-periodicity, we study the asymptotic behaviour of the complete Markovian process, showing that inside each of the groups mentioned above the distribution approaches a common normal distribution inx-space, with mean value and variance proportional tot. Explicit expressions for the proportionality factors are derived.The general theory is applied to the electrodiffusion equations, corresponding ton=2.