A Does there exist a proof for these conjectures?

[hilbert2]

TL;DR Summary

Made up some unproven(?) statements about simple one particle QM.

Does anyone know if a proof exists for these statements about 1d quantum mechanics?

1. If the potential energy where a particle moves is of the form

$V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots$

or

$V(x) = c_2 x^2 + c_3 |x|^3 + c_4 x^4 + c_5 |x|^5 + c_6 x^6 + \dots$

with $c_j \geq 0$ for all $j\in\mathbb{N}$, then the standard deviation of the particle's position in the eigenstates of $\hat{H}$ increases monotonically with increasing quantum number.

2. If the $V(x)$ is like that above, then the energy level spacing $E_{n+1} - E_n$ can only increase or stay constant with increasing quantum number $n$.

3. A system having any spectrum like $E_n = a + bn^c$ with $a,b,c$ constants, $1\leq c \leq 2$ and $b>0$ can be produced with a $V(x)$ of the form given above.

The claim number 2 seems to be correct, because if you begin with the harmonic oscillator potential $V(x) = \frac{1}{2}kx^2$ and add a small perturbation of form $V'(x) = \lambda x^{2n}$ with $n\in\mathbb{N}$ and $n\geq 2$, then the first order correction to the $m$:th energy eigenvalue is $E_{m}^{'} = \lambda\int\limits_{x=-\infty}^{x=\infty}\psi_{m}^{*}(x)x^{2n}\psi_m (x)dx$. This clearly gets larger with increasing $m$ (basically because of what was said in claim 1, i.e. the particle is more likely to be far from the origin $x=0$ for higher excited states), so this perturbation will only increase energy level spacings. But it's not rigorously clear that one can assume claim 1 is true and the first-order change can be used to predict the result of finite perturbations.

I invented claim 3 because of the observation that a spectrum like $E_n = a + bn^c$ usually fits very accurately to the first 10 - 20 eigenenergies of a system with a convex potential energy function $V(x)$. Only with larger quantum numbers it's easy to see the difference. So the claim 3 is the converse of that statement. However, the potential energy $V(x) = c_2 x^2 + c_4 x^4 + c_6 x^6 + \dots$ is not necessarily the only one that produces the same spectrum - a system with constant energy level spacings can also be created with the singular "isotonic oscillator potential" ( https://www.sciencedirect.com/science/article/abs/pii/037596017990197X ) instead of a harmonic oscillator. One way to reconstruct a $V(x)$ corresponding to a given spectrum $E_n$ is the inverse scattering theorem, but it's not necessarily easy to calculate the limit of Eqn. (5a) in the link https://arxiv.org/pdf/0811.1389.pdf when all eigenvalues are included and the determinant becomes "infinitely large".

 

[Haborix]

Regarding #2, I recall at some point looking at work on Sturm-Liouville problems for the asymptotics of eigenvalues at large quantum numbers. But I also remember everything was much more complicated when the domain was the whole line as opposed to some compact segment.

 

[hilbert2]

Regarding #2, I recall at some point looking at work on Sturm-Liouville problems for the asymptotics of eigenvalues at large quantum numbers. But I also remember everything was much more complicated when the domain was the whole line as opposed to some compact segment.

That sounds like it has something to do with the WKB approximation... But if the system is on a compact segment of the real line, then shouldn't the eigenvalues behave like $E_n \propto n^2$ for large $n$, as in the particle in a box problem?

When trying to fit those functions $E_n = a + bn^c$ to spectra of systems where the graph of $V(x)$ can only turn upward, the clearest difference was already seen in the low eigenvalues when the potential energy was $V(x) = C|x|$ for $|x| < L/2$ and $V(x) = \infty$ for $|x| \geq L/2$ for some well diameter $L$. In this potential, the particle should "see" a linear slope at low energies and only care about the hard walls at $|x|=L/2$ with high energy eigenstates. In that one the two different "extreme cases" are found from the same system.