I am wondering how (qualitatively) the energies of the boundstates in an infinite square well with a small potential step in the middle changes if we change that potential step. The problem is actually pretty similar to  [this post], but I especially would like to know how the gaps between the energies changes, if we change the height or the width of the potential step.

I tried to find a formula for the energies by solving the time independent problem, but couldn't find a clear solution (it looked more like a transcendantal equation but I might have made some mistakes).

  [this post]: https://physics.stackexchange.com/questions/40804/barrier-in-an-infinite-double-well/40813