Reviews in Mathematical Physics

We study σ-additivity of the physically plausible Jauch–Piron states on a von Neumann algebra M. Amongst other consequences we extend our earlier results [5] by showing that geometric conditions much weaker than pureness imply σ-additivity for a Jauch–Piron state and, further, that if M is properly infinite and the continuum hypothesis is assumed to be true then all Jauch–Piron factor states are σ-additive.

For the low temperature Blume–Emery–Griffiths Z^d, d ≥2, lattice model taking site spin values 0, +1, -1 we construct, using a polymer expansion, two pure states in the parameter region where there are an infinite number of configurations with minimal energy. Each state is invariant under translation by two lattice spacings and the two states are related by a unit translation. Using analyticity techniques we show that the truncated n-point function decays exponentially with an n-independent lower bound on the decay rate. For the truncated two-point function, we find the exact exponential decay rate in the limit β→∞.

In this article we continue our analysis of Schrödinger operators with a random potential using scattering theory. In particular the theory of Krein's spectral shift function leads to an alternative construction of the density of states in arbitrary dimensions. For arbitrary dimension we show existence of the spectral shift density, which is defined as the bulk limit of the spectral shift function per unit interaction volume. This density equals the difference of the density of states for the free and the interaction theory. This extends the results previously obtained by the authors in one dimension. Also we consider the case where the interaction is concentrated near a hyperplane.

The geometrical diffraction theory, in the sense of Keller, is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the "diffracted rays", which follow from the non-uniqueness of the Cauchy problem for geodesics in a Riemannian manifold with boundary. Then, the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian geometry, and the results of the Morse theory can be applied. The methods of the algebraic topology allow us to introduce the homotopy classes of diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope of the diffracted rays and, instead of the standard saddle point method, use is made of the procedure of Chester, Friedman and Ursell to derive the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated.

A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q²=e^-iℏ. The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation