# Is there some quantum potential producing exponential eigenvalues?

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?

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 $n^2$.
• Guessing... a finite set of bound states growing exponentially could approximate some potential of "tidal" kind, $V(x) \sim 1/x^2$, couldn't it? – arivero Jul 2 '14 at 12:27
• A negative attractive 1D potential of the asymptotic form $$-V(x) \sim 1/x^2\quad\text{for}\quad |x|\to \infty$$ (where we implicitly assume that a singularity at $x=0$ has been regularized) would lead to exponentially decaying energy levels $$-E_n \sim e^{-\mu n}\quad\text{for}\quad n\to \infty.$$ This potential is also discussed in this Phys.SE answer. – Qmechanic Jul 2 '14 at 12:42