We consider the focusing energy-critical nonlinear SchrödingerW 2 , then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.

Some algebraic aspects of field quantization in space-time with boundaries are discussed. We introduce an associative algebra B R , whose exchange properties are inferred from the scattering processes in integrable models with reflecting boundary conditions on the half line. The basic properties of B R are established and the Fock representations associated with certain involutions in B R are derived. We apply these results for the construction of quantum fields and for the study of scattering on the half line.

We obtain and analyze an exact solution to Einstein-Maxwell-scalar theory in (2+ 1) dimensions, in which the scalar field couples to gravity in a nonminimal way, and it also couples to itself with the self-interacting potential solely determined by the metric ansatz. A negative cosmological constant naturally emerges as a constant term in the scalar potential. The metric is static and circularly symmetric and contains a curvature singularity at the origin. The conditions for the metric to contain 0, 1 and 2 horizons are identified, and the effects of the scalar and electric charges on the size of the black hole radius are discussed. Under proper choices of parameters, the metric degenerates into some previously known solutions in (2 + 1)-dimensional gravity.

The thermodynamic phase space of GaussBonnet (GB) AdS black holes is extended, taking the inverse of the GB coupling constant as a new thermodynamic pressure P GB . We studied the critical behavior associated with P GB in the extended thermodynamic phase space at fixed cosmological constant and electric charge. The result shows that when the black holes are neutral, the associated critical points can only exist in five dimensional GB-AdS black holes with spherical topology, and the corresponding critical exponents are identical to those for the Van der Waals system. For charged GB-AdS black holes, it is shown that there can be only one critical point in five dimensions (for black holes with either spherical or hyperbolic topologies), which also requires the electric charge to be bounded within some appropriate range; while in d > 5 dimensions, there can be up to two different critical points at the same electric charge, and the phase transition can occur only at temperatures which are not in between the two critical values.