In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previous ones to validate a Bondi-type mass loss formula. The Lorentz group acts on the asymptotic infinity through the exchange of foliations by coordinate spheres. For foliations aligning with the total energy-momentum vector, we prove that the limits of quasi-local center of mass and angular momentum are finite, and evaluate the limits in terms of the expansion coefficients of the metric and the second fundamental form.

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We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |$q$($x$)|≤$C$(1+|$x$|)^{−1/4−ε}; in the smoothness direction, a sufficient condition in Hölder classes is$q$∈$C$^{1/2+ε}($R$). On the other hand, we show that there exist potentials which both satisfy |$q$($x$)|≤$C$(1+|$x$|)^{−1/4}and belong to$C$^{1/2}($R$) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.

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