Phase separation and collective excitations of two-component Bose-Einsteincondensates in traps

The phase separation and collective excitations of two-component Bose-Einst ein condensates in traps are studied. Using the variational method, we show that the separated state is favorable in the case where the interspecies i nteraction is strongly repulsive and the number of atoms is large enough. T his phase separation condition is explained in terms of the softening of a dipole mode collective excitation. Other low-lying excitation modes and the ir softenings which lead to the instability of the mixed state are also stu died by use of the sum-rule approach.