Note to the paper by R. Hempel, A. M. Hinz, and H. Kalf: On the essential spectrum of Schr�dinger operators with spherically symmetric potentials

1987 ◽  
Vol 277 (2) ◽  
pp. 209-211 ◽  
Author(s):  
J. Weidmann
Keyword(s):  
Essential Spectrum ◽  
Spherically Symmetric

On the essential spectrum of Schr�dinger operators with spherically symmetric potentials

1987 ◽  
Vol 277 (2) ◽  
pp. 197-208 ◽  
Author(s):  
Rainer Hempel ◽  
Andreas M. Hinz ◽  
Hubert Kalf
Keyword(s):  
Essential Spectrum ◽  
Spherically Symmetric

scholarly journals On the Existence of Linearly Oscillating Galaxies

2021 ◽  
Author(s):  
Mahir Hadžić ◽  
Gerhard Rein ◽  
Christopher Straub
Keyword(s):  
Steady State ◽  
Essential Spectrum ◽  
Adjoint Operator ◽  
Steady States ◽  
Symmetric Case ◽  
Spherically Symmetric ◽  
General Criterion ◽  
Period Function ◽  
Plane Symmetric ◽  
Monotonicity Assumption

AbstractWe consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

New branches of the essential spectrum of a 2 × 2 operator matrix

2020 ◽  
Vol 2020 (2) ◽  
pp. 44-51
Author(s):  
E.B. Dilmurodov
Keyword(s):  
Essential Spectrum ◽  
Operator Matrix

The two-body problem: an effective-one-body approach

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Reaction Force ◽  
Kerr Black Hole ◽  
Radiation Reaction ◽  
Spherically Symmetric ◽  
Geodesic Motion ◽  
Two Body Problem ◽  
Symmetric Spacetime ◽  
Radiation Reaction Force

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.

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