differential equations - Does there exist a potential which realizes this strange quantum mechanical system?

I have done some courses on quantum mechanics and statistical mechanics in the past. Since I also do math, I wonder about converge issues which are usually not such a problem in physics. One of those questions is the following. I will describe the background, but in the end it boils down to a question about ordinary differential equations.

In quantum mechanics on the real line, we start with a potential $V: mathbb{R} to mathbb{R}$ and try to solve the Schrödinger question $ihbar frac{partial}{partial t}Psi(x,t) = - frac{hbar^2}{2m}frac{partial^2}{partial x^2}Psi(x,t)+V(x)Psi(x,t)$. In many cases this can be accomplished by seperating variables, in which case we obtain the equation $EPsi(x,t) = - frac{hbar^2}{2m}frac{partial^2}{partial x^2}Psi(x,t)+V(x)Psi(x,t)$ which we try to solve for $E$ and $Psi$ to obtain a basis for our space of states together with an associated energy spectrum. For example, if we have a harmonic oscillator, $V(x) = frac{1}{2}momega^2x^2$ and we get $E_n = hbar omega (n+frac{1}{2})$ and $Psi_n$ a certain product of exponentials and Hermite polynomials. We assume that the energy in normalized such that the lowest energy state has energy $0$.

If the states of our system are non-degenerate, i.e. there is only one state for each energy level in the spectrum, then the partition function in statistical mechanics for this system is given by the sum $Z(beta) = sum_n exp(-beta E_n)$, where $beta$ is the inverse temperature $frac{1}{k_B T}$. It is clear that this sum can be divergent; in fact for a free particle ($V = 0$), it is not even well defined since spectrum is a continuum.

However, I was wondering about the following question: Is there a system such that $Z(beta)$ diverges for $beta < alpha$ and converges for $beta > alpha$ for some $alpha in mathbb{R}_{> 0}$? Am I correct in thinking that such a system is most likely an approximation of another system, which undergoes a phase transition at $beta = alpha$?

Anyway, an obvious candidate would be a potential $V$ such that the spectrum is given $E_n = C log(n+1)$ for $n geq 0$ and $C > 0$. This gets me to my main mathematical question: Does such a potential (or one with spectrum asymptotically similar) exist? If so, can you give it explicitly?

One the circle, the theory of Sturm-Liouville equations tells us that the eigenvalues must go asymptotically as $C n^2$, so in this case such problems can't occur. I don't know much about spectral theory for Sturm-Liouville equations on the real line though. The second question is therefore: What is known about the asymptotics of the spectrum of a Sturm-Liouville operator on the real line?

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