The moment inequalities due to Lieb and Thirring are effective tools in the
operator theory. Especially the one for the sum of the (negative)
eigenvalues of a Schrödinger operator, since, by duality, it is equivalent
to a lower bound for the kinetic energy of Fermions, which is exactly of the
right semi-classical Thomas-Fermi type.
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.
