Example Exact Solutions of the Time-independent Gross-Pitaevskii and Schr\"odinger Equations

A prescription is given to obtain some exact results for certain external potentials $V\left(\vec{r}\right)$ of the time-independent Gross-Pitaevskii and Schr"odinger equations. The study motivation is the ability to program $V\left(\vec{r}\right)$ experimentally in Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given $V\left({\vec r}\right)$,weaskwhich$V\left({\vec r}\right)$ will have a given pdf (probability density function) $P\left(\vec{r}\right)$. Several examples in 1D and 2D are presented for well-known pdfs and for the hydrogen atom in momentum space.