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We study the asymptotic behavior of the counting function of negative eigenvalues of Schrödinger operators with real valued potentials which decay at infinity on asymptotically hyperbolic manifolds. We establish conditions on the rate of decay of the potential that determine if there are finitely or infinitely many negative eigenvalues. In the latter case, they may only accumulate at zero and we obtain the asymptotic behavior of the counting function of eigenvalues in an interval(-\infty, -E)asE\rightarrow 0. 

Abstract Thermospheric density influences the atmospheric drag and is crucial for space missions. This paper introduces a global thermospheric density prediction framework based on a deep evidential method. The proposed framework predicts thermospheric density at the required time and geographic position with given geomagnetic and solar indices. It is called global to differentiate it from existing research that only predicts density along a satellite orbit. Through the deep evidential method, we assimilate data from various sources including solar and geomagnetic conditions, accelerometer‐derived density data, and empirical models including the Jacchia‐Bowman model (JB‐2008) and the Naval Research Laboratory Mass Spectrometer and Incoherent Scatter Radar Extended (NRLMSISE‐00) model. The framework is investigated on five test cases along various satellites from 2003 to 2015 involving geomagnetic storms with Disturbance Storm Time (Dst) values smaller than −50 . Results show that the proposed framework can generate density with higher accuracy than the two empirical models. It can also obtain reliable uncertainty estimations. Global density estimations at altitudes from 200 to 550 km are also presented and compared with empirical models on both quiet and storm conditions.