We construct the boundary phase space in D-dimensional Einstein gravity with a generic given co-dimension one null surface N as the boundary. The associated boundary symmetry algebra is a semi-direct sum of diffeomorphisms of \mathcal{N} and Weyl rescalings. It is generated by D towers of surface charges that are generic functions over \mathcal{N}. These surface charges can be rendered integrable for appropriate slicings of the phase space, provided there is no graviton flux through \mathcal{N}. In one particular slicing of this type, the charge algebra is the direct sum of the Heisenberg algebra and diffeomorphisms of the transverse space, \mathcal{N}_v for any fixed value of the advanced time v. Finally, we introduce null surface expansion- and spin-memories, and discuss associated memory effects that encode the passage of gravitational waves through \mathcal{N}, imprinted in a change of the surface charges.

Consider an unstable quantum system that has been found undecayed at an instant s and denote by R(t,s) the rate of its regeneration into an original (undecayed) state at a later instant t+s. It is proved that the reduced evolution is a semigroup, i.e., there is no regeneration at all, provided R(t,s)1/2 can be estimated by a sufficiently regular function that is nondecreasing in s and has zero derivative with respect to t at t=0 for every s. This generalizes the theorem of Misra and Sinha [Helv. Phys. Acta 45, 619 (1972)] in a different direction than in a recent paper by Nishioka [J. Math. Phys. 29, 1860 (1988)].

We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.

The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F₁-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.