Weakly interacting atomic or molecular bosons in quantum degenerate regimeand trapped in harmonically confined optical lattices, exhibit a wedding cakestructure consisting of insulating (Mott) shells. It is shown that superfluidregions emerge between Mott shells as a result of fluctuations due to finitehopping. It is found that the order parameter equation in the superfluidregions is not of the Gross-Pitaeviskii type except near the insulator tosuperfluid boundaries. The excitation spectra in the Mott and superfluidregions are obtained, and it is shown that the superfluid shells posses lowenergy sound modes with spatially dependent sound velocity described by a localindex of refraction directly related to the local superfluid density. Lastly,the Berezinskii-Kosterlitz-Thouless transition and vortex-antivortex pairs arediscussed in thin (wide) superfluid shells (rings) limited by three (two)dimensional Mott regions.
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