A parabolic model for dimple potentials

We study the 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 Schrodinger equation, has the same bound state spectrum, tunneling and reflection amplitudes as 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 a harmonic trap with a delta function at the origin is a limiting case of the harmonic trap with a symmetric truncated parabolic potential around the origin. Hence, the truncated parabolic is a better candidate for modeling the dimple potentials.