On some states minimizing uncertainty relations

Analyzing Heisenberg-Robertson (HR) and Schr\"{o}dinger uncertainty relations we found, that there can exist large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of non-commuting observables, $A$ and $B$, is zero. These states are not eigenstates of either the observable $A$ or $B$. We have also shown that the so-called "sum uncertainty relations" also do not provide any information about lower bounds on the standard deviations calculated for these states.
Comment: 12 pages, Neww comments and results added