The double beta decays of ${}^{76}\mathrm{Ge}$, ${}^{82}\mathrm{Se}$, and ${}^{136}\mathrm{Xe}$ are studied in very large shell model spaces. The dimensions of the shell model spaces used in these calculations reach $O\left({10}^8\right)$. The two neutrino matrix elements obtained are in good agreement with the available experimental data. Assuming that the mass of the neutrino is smaller than 1 eV, we get the following upper bounds to the half-lives for the neutrinoless decays: $T_{1/2}^{\left(0\nu \right)}\left(\mathrm{Ge}\right)>1.85\mathrm{}\times {10}^{25}\mathrm{yr}$, $T_{1/2}^{\left(0\nu \right)}\left(\mathrm{Se}\right)>2.36\mathrm{}\times {10}^{24}\mathrm{yr}$, and $T_{1/2}^{\left(0\nu \right)}\left(\mathrm{Xe}\right)>1.21\mathrm{}\times 10^{25}\mathrm{yr}$.

• Received 10 January 1996

DOI:https://doi.org/10.1103/PhysRevLett.77.1954