Stability of Atoms StdQM vs RealQM vs Lieb-Thirring

This is a clarification of previous posts on Stability of Matter connecting to the complex mathematical analysis of Lieb-Thirring-Dyson-Lenard. 

We noticed that the total energy $E(\Psi )$ of any admissible wave function $\Psi (x)$ for the Hydrogen atom with $Z=N=1$ satisfies 

• $E(\Psi )\ge -\frac{\pi^2}{8}\approx -1.23$  

as a consequence of the Kato-Hardy inequality

• $\int\frac{\Psi^2 (x)}{\vert x\vert}dx \le \frac{\pi}{2}\int\vert\nabla\Psi (x)\vert^2dx$

for all $\Psi\in H^1(R^3)$. 

The proof directly extends to an atom of charge $Z>1$ with $N=Z$ electrons described by StdQM in the form 

• $E(\Psi )\ge -CNZ^2 = -CZ^3$

with some positive constant $C$, which is way too small allowing all electrons to concentrate around the kernel without any shell structure.

With the atom described by RealQM we get instead using shell structure (see this post)

• $E(\Psi )\ge- CZ^2$

which fits with observations. 

We see that the stability of an atom as total energy bounded below, is easy to prove mathematically using the Kato-Hardy inequality (with simple proof recalled in this post). The proof in the case of RealQM also gives the dependence on $Z$ observed. 

The question is what Lieb-Thirring-Dyson-Lenard adds to this clear picture. In this work the Lieb-Thirring inequality

• $\int\vert\nabla\Psi (x)\vert^2dx \ge C\int\rho^{\frac{5}{3}}(x)dx$ 

where $\rho (x)$ is a joint electron density, plays a central role. But it means passing to a weaker control of potential energy and so makes stability harder to prove, seemingly without pressing reason.