It is a long standing open question, whether the semiclassical limit for the energy of fermions coincides with the lower bound and a lot of work has been done in the last 20 years to improve the constants in these bounds.

Based on ideas of Rumin, we show a novel approach of proving the Lieb-Thirring inequalities for the operator $H=|p|^k-U$ with arbitrary $k>0$ in any dimension $d$. The obtained constants are improvements of currently known results in all cases, in particular, for $k=2$.

The other advantage is that the derived factors relating our inequality to semiclassical ones, that is, the quotient of our constants divided by the semi-classical guess, are uniformly bounded for all $k$ and $d$ by $e$.

We also estimate number of negative eigenvalues for the operator $H$ with dimension $d>k$. Factoring out the semiclassical estimate on the number of bound states yields a uniformly bounded estimate converging to $e^2$ for large dimensions. These results work for all $k$ and do not use an extension of the bounds to operator-valued potentials and the induction in the dimension trick of Laptev and Weidl, which works only for $k=2$.

This seems to be the first time that one can prove universal bounds without using some type of induction in the dimension argument. However, for $k=2$ one can do this and we get bounds which improving the bounds for small values of $d$ in this case.