Longitudinal Profiles of Atom Laser Propagation in a Cigar-Shaped Trap

Focusing on the cigar-shaped trap model, we provide longitudinal profiles of a weakly outcoupled atom laser propagation both inside and outside the Bose-Einstein condensate regions. The propagation itself is generally represented by inhomogeneous Schrödinger equation which is derived from a set of Gross-Pitaevskii equations by applying the available conditions. We also show that by imposing boundary condition and using quantum oscillator model, energy of the outcoupled atom laser outside the Bose-Einstein condensate region is quantized while there is no analytical solution for the propagation of the outcoupled atom laser inside the Bose-Einstein condensate region.

SOME RESULTS ON INVERSE SCATTERING

A review of some of the author's results in the area of inverse scattering is given. The following topics are discussed: (1) Property C and applications, (2) Stable inversion of fixed-energy 3D scattering data and its error estimate, (3) Inverse scattering with “incomplete” data, (4) Inverse scattering for inhomogeneous Schrödinger equation, (5) Krein's inverse scattering method, (6) Invertibility of the steps in Gel'fand–Levitan, Marchenko, and Krein inversion methods, (7) The Newton–Sabatier and Cox–Thompson procedures are not inversion methods, (8) Resonances: existence, location, perturbation theory, 9) Born inversion as an ill-posed problem, (10) Inverse obstacle scattering with fixed-frequency data, (11) Inverse scattering with data at a fixed energy and a fixed incident direction, (12) Creating materials with a desired refraction coefficient and wave-focusing properties.

AN ABSTRACT RADIATION CONDITION AND APPLICATIONS TO N-BODY SYSTEMS

In the setting of Mourre [18] we characterize the "outgoing" and "incoming" solutions, to the abstract inhomogeneous Schrödinger equation (H-E)u=v, given by the Limiting Absorption Principle. The characterization is in terms of an abstract radiation condition and as an application we give a characterization, in the framework of weighted L2-spaces, of the outgoing and incoming solutions for N-body Schrödinger operators with and without Stark effect. The abstract radiation condition translates in the application into a radiation condition considered by Isozaki in [13].