So I have a strange quantum potential I have been playing with:
$$V(x) = \frac{x^2}{5}+\mu \left(\left\lfloor x+\frac{1}{2}\right\rfloor \right).$$
where $\mu$ is the Möbius function. This is what it looks like.
I wanted to see what the bound states looked like, but $V$ is discontinuous, so I changed it into the following:
$$\frac{x^2}{5}+\sum _{n=-M}^M\mu(n)\left[\tanh\left(b\left(x-\frac{1}{2}-n+1)\right)\right)-\tanh\left(b\left(x+\frac{1}{2}-n-1\right)\right)\right]$$
This $\tanh$ method is a common way to approximate step functions in numerical quantum mechanics. The limit of the above expression as $M$ and $b$ go to infinity is $V$.
I picked a reasonably high value for $b$ and took $M$ out to $10$. The bound states ended up looking like this: (LHS is $\psi_n$, RHS is $|\psi_n|$ overlayed on $V$)
and so on. I don't have anything much smarter to say about these than that they are pretty weird looking.
Can anything rigorous be said about the eigenfunctions of this potential?
By this, I mean the potential in the limit as $M,b\to\infty$.
Mainly I am wondering whether including higher $M$ would have given me more bound states between the above energies. The energies seem to be at very strange intervals, like there could perhaps should have been more between them. In particular, if I were to take $M$ out to infinity, would I get a dense spectrum of energies?