Analytical Solution for the Deformation of a Cylinder under Tidal Gravitational Forces

Quite a few future high precision space missions for testing Special and General Relativity will use optical resonators which are used for laser frequency stabilization. These devices are used for carrying out tests of the isotropy of light (Michelson-Morley experiment) and of the universality of the gravitational redshift. As the resonator frequency not only depends on the speed of light but also on the resonator length, the quality of these measurements is very sensitive to elastic deformations of the optical resonator itself. As a consequence, a detailed knowledge about the deformations of the cavity is necessary. Therefore in this article we investigate the modeling of optical resonators in a space environment. Usually for simulation issues the Finite Element Method (FEM) is applied in order to investigate the influence of disturbances on the resonator measurements. However, for a careful control of the numerical quality of FEM simulations a comparison with an analytical solution of a simplified resonator model is beneficial. In this article we present an analytical solution for the problem of an elastic, isotropic, homogeneous free-flying cylinder in space under the influence of a tidal gravitational force. The solution is gained by solving the linear equations of elasticity for special boundary conditions. The applicability of using FEM codes for these simulations shall be verified through the comparison of the analytical solution with the results gained within the FEM code.Comment: 23 pages, 3 figure

Construction of analytic, unitary scattering amplitudes from a given differential cross-section:A refined analysis

We extend previous work concerning the construction of unitary scattering amplitudes that correspond to the scattering data at a given energy. The dispersive and absorptive parts are by construction analytic in cosϑ in the small and large Lehmann ellipses, respectively. The dispersive and absorptive parts obtained here, in contrast to those obtained before, are shown to have continuous derivatives on the boundary of their domains of analyticity. The continuum ambiguity in the determination of the scattering amplitude, which is associated with a lack of experimental information on the inelastic contribution to unitarity, is present here as well

Monotonie und Randspektrum bei vollstetigen Operatoren

Elsner L. Monotonie und Randspektrum bei vollstetigen Operatoren. Archive for Rational Mechanics and Analysis. 1970;36(5):356-365.Es sei A ein vollstetiger linearer Operator eines Banachraums in sich. Es wird gezeigt, daß sich die Eigenschaft ldquores gibt eine Halbordnung mit totalem Ordnungskegel, bezüglich der A monoton ist rdquo durch Aussagen allein über diejenigen Eigenwerte von A, deren Betrag gleich dem Spektralradius ist, charakterisieren läßt. Ähnliche Äquivalenzaussagen gelten für irreduzible und streng-monotone Operatoren