Abstract: Starting from a short review on some classical spectral estimates,
in particular bounds on the spectral data by the corresponding semi-
classical phase space volume such as the Li-Yau and the Lieb-Thirring
inequalities, I will discuss in this survey talk a number of open problems
and conjectures in this area.
On the one hand these are classical unresolved conjectures,
on the other hand even slight and natural changes of the classical
statement of the problem lead to unresolved questions.