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Instabilities for boson-fermion mixed condensates of trapped Alkali atoms dueto the boson-fermion attractive interaction are studied using a variationalmethod. Three regions are shown for their instabilities according to theboson-fermion interaction strength: stable, meta-stable and unstable ones. Thestability condition is obtained analytically from the asymptotic expansion ofthe variational total energy. The life-time of metastable states is discussedfor tunneling decay, and is estimated to be very long. It suggests that, exceptnear the unstable border, meta-stable mixed condensate should be almost-stableagainst clusterizations. The critical border between meta-stable and unstablephases is calculated numerically and is shown to be consistent with theM{\o}lmer scaling condition.Comment: 7 pages, 3figure

Induced instability for boson-fermion mixed condensates of alkali-metal atoms due to the attractive boson-fermion interaction