Spectral properties of a model potential for quantum dots with smooth boundaries

A model potential for quantum dots with smooth boundaries that is qualitatively realistic in the whole energy range is presented. Starting from exact zero energy eigenfunctions, we characterize for this potential the whole discrete spectrum, from the continuum threshold down to the harmonic-oscillator regime on the well bottom. The presented potential is singled out for being linked with the linear spectral problem associated to the $n$ soliton solution of the KdV equation.