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chemical potential. The figure shows the constant eigenfrequency of the dipolar mode [see straight (blue) line] at the value of the trap frequency ω$_{z}$ and the anomalous mode bifurcating from the dipolar mode in the linear limit. As mentioned above, the eigenfrequency of the anomalous mode reaches asymptotically the value ω$_{ds}$ = 30 . 36 Hz. 

The second panel of Fig. 4 shows the BdG spectrum for V$_{0}$ = 8 . 3 × 10 − 14 eV. All eigen- frequencies are real, so the state is still stable. The dashed line is the prediction by Eq. (21) using ω$_{ds}$ = ω$_{z}$$_{/}$√$_{2, while the solid line is the prediction with}$ ω$_{ds}$ = 30 . 36 Hz. The dipole mode remains approximately constant for such a small optical lattice strength. However, the degeneracy of the anomalous mode and the dipole mode in the linear limit disappears. 

The third panel shows the spectra for V$_{0}$ = 1 . 65 × 10 − 13 eV. In this case, the anomalous mode becomes purely imaginary for increasing chemical potential, reflecting the fact that the state becomes unstable. Notice that the numerical result and the analytical prediction, ω$_{ODE}$ = i √ | ω 2 ds − k $^{2}$V$_{0}$ | , are in a good agreement for large values of the chemical potential, where the TF limit is reached and Eq. (19) is valid. Since V$_{0}$ > V$_{0}$$_{,cr}$ , this state is expected to be unstable in the TF limit. 

The fourth panel shows the spectra for V$_{0}$ = 4 . 96 × 10 − 13 eV. In this case, the anomalous mode becomes purely imaginary already at small chemical potentials. Furthermore, the 

