  Linear response of simple (i.e., condensed) Bose-Einstein condensates is
known to lead to the Bogoliubov- de Gennes equations. Here, we derive linear
response for fragmented Bose-Einstein condensates, i.e., for the case where the
many-body wave function is not a product of one, but of several single-particle
states (orbitals). Our approach is based on the number-conserving variational
time-dependent mean field theory [O. E. Alon, A. I. Streltsov, and L. S.
Cederbaum, Phys. Lett. A 362, 453 (2007)], which describes the time evolution
of best-mean field states. Correspondingly, we call our linear response theory
for fragmented states LR-BMF. In the derivation it follows naturally that
excitations are orthogonal to the ground-state orbitals. As applications
excitation spectra of Bose-Einstein condensates in double-well potentials are
calculated. Both symmetric and asymmetric double-wells are studied for several
interaction strengths and barrier heights. The cases of condensed and two-fold
fragmented ground states are compared. Interestingly, even in such situations
where the response frequencies of the two cases are computed to be close to
each other, which is the situation for the excitations well below the barrier,
striking differences in the density response in momentum space are found. For
excitations with an energy of the order of the barrier height, both the
energies and the density response of condensed and fragmented systems are very
different. In fragmented systems there is a class of "swapped" excitations
where an atom is transfered to the neighboring well. The mechanism of its
origin is discussed. In asymmetric wells, the response of a fragmented system
is purely local (i.e., finite in either one or the other well) with different
frequencies for the left and right fragments.
