Spherical Fourier Transforms on Locally Compact Quantum Gelfand Pairs

We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in $SL(2,\mathbb{C}$) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a $\mathbb{Z}_2$-grading, we obtain product formulae for little $q$-Jacobi functions

Asymptotic equality of the isolated and the adiabatic susceptibility

Many-particle systems with a hamiltonian of the form H = A + hB, h being a parameter, are discussed. In particular, for a certain class of these systems, a criterion is derived for the asymptotic equality of the isolated and the adiabatic susceptibility or, equivalently, for the ergodicity of B. This criterion states that, for sufficiently large particle number, any hermitian operator polynomial in h of any degree J that commutes with H(h) can be written as a linear combination of the powers H0, …, HJ with polynomial coefficients

Some exact excited states in a linear antiferromagnetic spin system

Exact expressions are derived for some excited states in a linear quantum spin system for which the exact ground state has been studied in the last decade

On the existence of a gap in the energy spectrum of quantum systems

A theorem on the existence of a gap in the energy spectrum of quantum systems, the exact ground state of which is known explicitly, is proved. The theorem is applied to a three-dimensional Heisenberg spin-1/2 ferromagnet, with anisotropic nearest-neighbour interactions, and to an alternating Heisenberg antiferromagnet, with nearest- and next-nearest-neighbour interactions

Elementary excitations in antiferromagnetic Heisenberg systems

The structure of the (eigen)states of antiferromagnetic Heisenberg systems is discussed. These systems are shown to be equivalent to classical systems of coupled harmonic oscillators. Most attention will be paid to the first excited state. This state is supposed to be a triplet. An approximation method, which is a generalization of a method, used to describe the ground state of Heisenberg systems, will be used to describe elementary excitations. The working of the method is demonstrated by some small-system calculations

Exact spectrum for n electrons in the single band Hubbard model

The energy spectrum and the correlation functions for n electrons in the one-dimensional single band Hubbard model with periodic boundary conditions are calculated exactly. For that purpose the Hamiltonian is transformed into a set of Hamiltonians, corresponding to systems of spinless fermions.\ud Our results include the results of Mei and Chen, presented in a recent paper

RF measurements I: signal receiving techniques

For the characterization of components, systems and signals in the RF and microwave range, several dedicated instruments are in use. In this paper the fundamentals of the RF-signal sampling technique, which has found widespread applications in 'digital' oscilloscopes and sampling scopes, are discussed. The key element in these front-ends is the Schottky diode which can be used either as an RF mixer or as a single sampler. The spectrum analyser has become an absolutely indispensable tool for RF signal analysis. Here the front-end is the RF mixer as the RF section of modern spectrum analysers has a rather complex architecture. The reasons for this complexity and certain working principles as well as limitations are discussed. In addition, an overview of the development of scalar and vector signal analysers is given. For the determination of the noise temperature of a one-port and the noise figure of a two-port, basic concepts and relations are shown. A brief discussion of commonly used noise measurement techniques concludes the paper.Comment: 24 pages, contribution to the CAS - CERN Accelerator School: Specialised Course on RF for Accelerators; 8 - 17 Jun 2010, Ebeltoft, Denmar