On the Completeness of the Quasinormal Modes of the Poeschl-Teller Potential

The completeness of the quasinormal modes of the wave equation with Poeschl-Teller potential is investigated. A main result is that after a large enough time $t_0$, the solutions of this equation corresponding to $C^{\infty}$-data with compact support can be expanded uniformly in time with respect to the quasinormal modes, thereby leading to absolutely convergent series. Explicit estimates for $t_0$ depending on both the support of the data and the point of observation are given. For the particular case of an ``early'' time and zero distance between the support of the data and observational point, it is shown that the corresponding series is not absolutely convergent, and hence that there is no associated sum which is independent of the order of summation.Comment: 22 pages, 2 figures, submitted to Comm. Math. Phy

Results on the spectrum of R-Modes of slowly rotating relativistic stars

The paper considers the spectrum of axial perturbations of slowly uniformly rotating general relativistic stars in the framework of Y. Kojima. In a first step towards a full analysis only the evolution equations are treated but not the constraint. Then it is found that the system is unstable due to a continuum of non real eigenvalues. In addition the resolvent of the associated generator of time evolution is found to have a special structure which was discussed in a previous paper. From this structure it follows the occurrence of a continuous part in the spectrum of oscillations at least if the system is restricted to a finite space as is done in most numerical investigations. Finally, it can be seen that higher order corrections in the rotation frequency can qualitatively influence the spectrum of the oscillations. As a consequence different descriptions of the star which are equivalent to first order could lead to different results with respect to the stability of the star

On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole

This short paper derives the constant of motion of a scalar field in the gravitational field of a Kerr black hole which is associated to a Killing tensor of that space-time. In addition, there is found a related new symmetry operator S for the solutions of the wave equation in that background. That operator is a partial differential operator with a leading order time derivative of the first order that commutes with a normal form of the wave operator. That form is obtained by multiplication of the wave operator from the left with the reciprocal of the coefficient function of its second order time derivative. It is shown that S induces an operator that commutes with the generator of time evolution in a formulation of the initial value problem for the wave equation in the setting of strongly continuous semigroups

Aggregation in Models with Quantity Constraints: The CES Aggregation Function

This paper is devoted to the problem of aggregation in models with quantity constraints. The focus is on quantity rationing macroeconomic (QRM) models where the micromarket outcome can be written as the minimum of several variables and where the diversity of situations across micromarkets is explicitly recognized. The aggregation result given in this paper generalizes that of Lambert (1988) to employment functions with more than two components, and leads to approximate aggregate functions of the CES variety. The approximation used can accomodate general variance-covariance structures. Simulation experiments show that the approximation error remains within reasonable bounds (1-4%). It thus seems that the CES formulation can accomodate a large variety of situations. It remains in particular valid when the (restrictive) conditions required to obtain the CES function as an exact result (independently identically distributed Weibull variables) are not satisfied.Macroeconomics; smoothing-by-aggregation; mismatch; approximation