A Parabolic Model for the Dimple Potentials

Abstract: We study truncated parabolic function and demonstrate that it is a representation of the Dirac delta function. We also show that the truncated parabolic function, used as a potential in the Schr\"{o}dinger equation, has the same bound state spectrum, tunneling and reflection amplitudes with the Dirac delta potential as the width of the parabola approximates to zero. Dirac delta potential is used to model dimple potentials which are utilized to increase the phase-space density of a Bose-Einstein condensate in a harmonic trap. We show that harmonic trap with a delta function at the origin is a limit case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Therefore, we propose that the truncated parabolic potential is a better candidate for modeling the dimple potentials.