We introduce the generalized Airy-Gauss (AiG) beams and analyze their propagation through optical systems described by ABCD matrices with complex elements in general. The transverse mathematical structure of the AiG beams is form-invariant under paraxial transformations. The conditions for square integrability of the beams are studied in detail. The AiG beam describes in a more realistic way the propagation of the Airy wave packets because AiG beams carry finite power, retain the nondiffracting propagation properties within a finite propagation distance, and can be realized experimentally to a very good approximation.
The squeezed state approach of the semiclassical limit of the time-dependent Schrödinger equationWe suggest an effective approach to separation of variables in the Schrijdinger equation with two space variables. Using it we classify inequivalent potentials V(xl ,x,) such that the corresponding Schrodinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrodinger equation with the anisotropic harmonic oscillator potential V= k ,x: + kg: and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schriidinger equation (V=O). 0 I995
We construct the general solutions of the system of nonlinear differential equations 2nu = 0, uµu µ = 0 in the four-and five-dimensional complex pseudo-Euclidean spaces. The results obtained are used to reduce the multi-dimensional nonlinear d'Alembert equation 24u = F (u) to ordinary differential equations and to construct its new exact solutions.
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