Publications Goldberg, M., and Visan, M. (2005) A counterexample to dispersive estimates for Schrödinger operators in higher dimensions. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4 . Abstract of the Dissertation We consider the initial value problem for the defocusing energy-critical nonlinear where the initial data u 0 is chosen to belong to the energy-spaceḢ 1 (R n ), and u(t, x) is a complex-valued function in spacetime R t

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... R n x . We use the induction on energy technique introduced by J. Bourgain and extend the frequency-localized interaction Morawetz inequality introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao in dimension n = 3 to higher dimensions. The results obtained are global well-posedness, scattering, and global L 2(n+2) n−2 t,x spacetime bounds for solutions to the initial value problem ( * ). x