Is there some quantum potential producing exponential eigenvalues?

Question

Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" potentials which have only a single discrete eigenvalue. I was wondering, what kind of potential do we need for producing an exponential $e^n$ set of discrete eigenvalues?

Answer 1

For 1D potentials, the sequence of bound state energy eigenvalues $E_n$ cannot grow faster than what happens in the case of an infinite well, i.e. $E_n$ cannot grow faster than $n^2$.