Exact eigenfunctions of the linear ramp potential in the Gross-Pitaevskii equation for the Bose-Einstein condensate

Properties of magnetically trapped Bose gases are investigated within the Gross-Pitaevskii approximation for the condensate wavefunction. A linear ramp potential in the one-dimensional representation is shown to be exactly solvable. The wavefunction takes the form of the second Painleve transcendent and can be very accurately estimated using elementary functions which are globally non-singular. We analyse the physical characteristics of these condensate wavefunctions whose novel feature is a damped oscillatory profile. The nodeless solution, which corresponds to the lowest energy state. agrees with its earlier estimate using a linear analysis, while the new damped oscillatory solutions reveal a spectrum of the condensate's excited, highly inhomogeneous excited states. (C) 2001 Elsevier Science B.V. All rights reserved.