The Helmholtz equation, also known as the reduced wave equation, has, in the case of complex eigenvalues, solutions which decay exponentially at infinity. The was proved by Kato fifty years ago. Potetials are allowed, provided they decay faster than the Coulomb potential. Kato's result is suboptimal in three ways: (1) his decay rate is not the best possible, (2) his estimates are in L2 rather than in sup norm, and (3) he is missing an interesting derivative estimate, closely related to the Sommerfeld radiation condition. I will show how to fix all of these problems.
One of the traditional geophysical tools that allow us to study the Earth's interior is the electromagnetic induction. Time-varying electric currents in the Earth's ionosphere and magnetosphere induce secondary currents in the crust and mantle. Observations of the geomagnetic field at the surface and, recently, also from the space, can be used to reconstruct the radial and even lateral distribution of electrical conductivity which is closely related to the temperature, chemical, and mineralogical composition of the Earth. Mathematically, this technique leads to a typical non-linear inversion, where uniqueness of the solution is not guaranteed.
Interstellar chemistry plays an important role in the evolution of the universe from the dark ages to the formation of life. I will discuss two particularly important links in this chain of chemical reactions. The first is the formation of H2 in the early universe. Uncertainties in this process limit our ability to reliably model the formation of the first stars. The other link we study is the start of the cosmic pathway to life which begins in interstellar gas clouds where atomic carbon is "fixed" into molecules, thereby initiating the synthesis of the complex organic molecules that are eventually sequestered on planets. These reactions initiate not only the formation of organic molecules in the cosmos, but also provide some of the first threads knitting together atoms and molecules into solid material. Such processes are critical for the eventual formation of planets and may determine a major component of the organic chemistry that is present on their young surfaces. I will briefly review these motivations for our research and explain how we reproduce the relevant reactions through laboratory experiments.
Abstract. The angular power spectrum (anisotropy spectrum) of the Cosmic Microwave Background (relict radiation) shows incredible similarity with X-ray or neutron scattering measured on non-crystalline materials. Until now this spectrum has emerged as an important tool helping to test cosmological models and theories of cosmic structure formation. In this contribution we undertake an attempt to interpret the anisotropy spectrum of relict radiation using an analogous approach as is generally used when a structure of a disordered material is studied using X-ray (or neutron) scattering. We expect that various approximations used during this study may help to find conditions under which relict radiation scattering can be exploited to find similar information as in the case of the analysis of X-ray scattering on disordered materials. As a consequence we hope that this analysis may reveal interesting information (distances between objects and-or particles, density of the matter) on the ordering of the prime-matter which is scattering the relict radiation.