Schr\"odinger evolution of the Hawking state

A Schr\"odinger-picture description of the evolving quantum state of Hawking radiation is given, based on an ADM decomposition using time slicings that smoothly cross the horizon. This treatment avoids requiring a role for trans-planckian modes, which can be viewed as artifacts of Hawking's original calculation, and also supports arguments that radiation from black holes is produced in a "quantum atmosphere" with thickness comparable to the horizon size, rather than microscopically far from it. Particularly explicit formulas are given for the two-dimensional analog of the Schwarzschild geometry. This analysis is expected to generalize to other black holes, and to cosmology. The resulting quantum evolution also provides important background for investigating corrections to the Hawking process, as are necessary for restoring unitary evolution of black hole decay.