Limited-Diffraction Solutions to Maxwell and to Schroedinger Equations

We have developed a new family of limited-diffraction electromagnetic X-shaped waves based on the scalar "X waves" discovered previously. These waves are diffraction-free in theory and particle-like (wave packets), in that they maintain their shape as they propagate to an infinite distance. The X waves possess (theoretically) infinitely extended arms, and --at least, the ones studied in this paper-- have an infinite total energy: therefore, they are not physically realizable. However, they can be truncated in both space and time and approximated by means of a finite aperture radiator so to get a large enough depth of interest (depth of field). In addition to the Maxwell equations, X wave solutions to the free Schroedinger equation are also obtained. Possible applications of these new waves are discussed. Finally, we discuss the appearance of the X-shaped solutions from the purely geometric point of view of the Special Relativity theory. [PACS nos. 03.40.Kf, 03.50.De, 41.20.Jb, 41.20.Bt, 42.25.Bs, 03.30.+p, 03.65.-w].