We discuss the spectral subspace perturbation problem for a selfadjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tan Θ Theorem. We also extend the Davis-Kahan tan 2Θ Theorem to the case of some unbounded perturbations. Mathematics Subject Classification (2000). Primary 47A55; Secondary 47B25.

We formulate exact integral boundary conditions for a solution of the time-dependent Schrödinger equation that describes an atom interacting, in the dipole approximation, with a laser pulse. These conditions are imposed on a surface ͑boundary͒ which is usually chosen at a finite ͑but sufficiently remote͒ distance from the atom where the motion of electrons can be assumed to be semiclassical. For the numerical integration of the Schrödinger equation, these boundary conditions may be used to replace mask functions and diffuse absorbing potentials applied at the edge of the integration grid. These latter are usually introduced in order to ͑approxi-mately͒ compensate for unphysical reflection which occurs at the boundary of a finite region if a zero-value condition is imposed there on the solution. The present method allows one to reduce significantly the size of the space domain needed for numerical integration. Considering the numerical solution for a one-dimensional model, we demonstrate the effectiveness of our approach in comparison with some other numerical methods.

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We discuss long-range corrections for the integral boundary condition ͑IBC͒ introduced in A. M. Ermolaev, I. V. Puzynin, A. V. Selin, and S. I. Vinitsky, Phys. Rev. A 60, 4831 ͑1999͒, in the case of the time-dependent Schrödinger equation with a long-range atomic potential. As in the work of Ermolaev et al. the laser-atom interaction is taken in the dipole approximation. The IBC techniques require the knowledge of the Green's function of the problem, beyond some surface remote from the atom. We consider the eikonal approximation ͑EA͒ for the Green's function in the asymptotic region and perform numerical tests on a one-dimensional problem with the soft Coulomb potential. We demonstrate that the account of long-range corrections, within the EA, allows us to reduce significantly the size of the space domain required for numerical integration and improves essentially on the accuracy of the computed spectral distribution for the ejected electrons.