Majorana neutrinos, neutrino mass spectrum, and the vertical bar < m >vertical bar similar to 10(-3) eV frontier in neutrinoless double beta decay

If future neutrino oscillation experiments show that the neutrino mass spectrum is with normal ordering, m(1) &lt; m(2) &lt; m(3), and the searches for neutrinoless double beta [(beta beta)(0 nu)] decay with sensitivity to values of the effective Majorana mass |&lt; m &gt;| greater than or similar to 10(-2) eV give negative results, the next frontier in the quest for (beta beta)(0 nu)-decay will correspond to |&lt; m &gt;| similar to 10(-3) eV. By assuming that massive neutrinos are Majorana particles and their exchange is the dominant mechanism generating (beta beta)(0 nu) - decay, we analyze the conditions under which |&lt; m &gt;|, in the case of three-neutrino mixing and a neutrino mass spectrum with normal ordering, would satisfy |&lt; m &gt;| &gt;= 0.001 eV. We consider the specific cases of (i) a normal hierarchical neutrino mass spectrum, (ii) a relatively small value of the CHOOZ angle theta(13), as well as (iii) the general case of a spectrum with normal ordering, a partial hierarchy, and a value of theta(13) close to the existing upper limit. We study the ranges of the lightest neutrino mass m(1) and/or of sin(2)theta(13) for which |&lt; m &gt;| &gt;= 0.001 eV and discuss the phenomenological implications of such scenarios. We provide also an estimate of |&lt; m &gt;| when the three-neutrino masses and the neutrino mixing originate from a neutrino mass term of the Majorana type for the (left-handed) flavor neutrinos and Sigma(3)(j)m(j)U(ej)(2) = 0, but there does not exist a symmetry which forbids the (beta beta)(0 nu)-decay.