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Register a new userAperiodic order and spherical diffraction, III: The shadow transform and the diffraction formula
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We define spherical diffraction measures for a wide class of weighted point sets in commutative spaces, i.e. proper homogeneous spaces associated with Gelfand pairs. In the case of the hyperbolic plane we can interpret the spherical diffraction measure as the Mellin transform of the auto-correlation distribution. We show that uniform regular model sets in commutative spaces have a pure point spherical diffraction measure. The atoms of this measure are located at the spherical automorphic spectrum of the underlying lattice, and the diffraction coefficients can be characterized abstractly in terms of the so-called shadow transform of the characteristic functions of the window. In the case of the Heisenberg group we can give explicit formulas for these diffraction coefficients in terms of Bessel and Laguerre functions.
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We study uniform and non-uniform model sets in arbitrary locally compact second countable (lcsc) groups, which provide a natural generalization of uniform model sets in locally compact abelian groups as defined by Meyer and used as mathematical models of quasi-crystals. We then define a notion of auto-correlation for subsets of finite local complexitiy in arbitrary lcsc groups, which generalizes Hofs classical definition beyond the class of amenable groups, and provide a formula for the auto-correlation of a regular model set. Along the way we show that the punctured hull of an arbitrary regular model set admits a unique invariant probability measure, even in the case where the punctured hull is non-compact and the group is non-amenable. In fact this measure is also the unique stationary measure with respect to any admissible probability measure.
We study the auto-correlation measures of invariant random point processes in the hyperbolic plane which arise from various classes of aperiodic Delone sets. More generally, we study auto-correlation measures for large classes of Delone sets in (and even translation bounded measures on) arbitrary locally compact homogeneous metric spaces. We then specialize to the case of weighted model sets, in which we are able to derive more concrete formulas for the auto-correlation. In the case of Riemannian symmetric spaces we also explain how the auto-correlation of a weighted model set in a Riemannian symmetric space can be identified with a (typically non-tempered) positive-definite distribution on $mathbb R^n$. This paves the way for a diffraction theory for such model sets, which will be discussed in the sequel to the present article.
The Gutzwiller trace formula is extended to include diffraction effects. The new trace formula involves periodic rays which have non-geometrical segments as a result of diffraction on the surfaces and edges of the scatter.
Surface electromagnetic modes supported by metal surfaces have a great potential for uses in miniaturised detectors and optical circuits. For many applications these modes are excited locally. In the optical regime, Surface Plasmon Polaritons (SPPs) have been thought to dominate the fields at the surface, beyond a transition region comprising 3-4 wavelengths from the source. In this work we demonstrate that at sufficiently long distances SPPs are not the main contribution to the field. Instead, for all metals, a different type of wave prevails, which we term Norton waves for their reminiscence to those found in the radio-wave regime at the surface of the Earth. Our results show that Norton Waves are stronger at the surface than SPPs at distances larger than 6-9 SPPs absorption lengths, the precise value depending on wavelength and metal. Moreover, Norton waves decay more slowly than SPPs in the direction normal to the surface.
In these lectures, we present and discuss the most recent results on inclusive diffraction at the Tevatron collider and give the prospects at the LHC. We also describe the search for exclusive events at the Tevatron. Of special interest is the exclusive production of Higgs boson and heavy objects ($W$, top, stop pairs) at the LHC which will require precise measurements and analyses of inclusive and exclusive diffraction to constrain further the gluon density in the pomeron. At the end of these lectures, we describe the projects to install forward detectors at the LHC to fulfil these measurements.
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